instructions: first separate the AmS TeX part (ending at "\enddocument") from the postscript part (the rest). The postscript file should be called fig1.ps. Then compile the AmS TeX file using TeX. If psfig version 1.9 is correctly installed, this should produce a complete version of the article. If the figure is not inserted correctly, print fig1.ps separately. BODY \magnification=\magstep1 \input amstex \input psfig \documentstyle{amsppt} \vsize=22 truecm \hsize=16 truecm \TagsOnRight \NoRunningHeads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def \real{{\Bbb R}} \def \complex{{\Bbb C}} \def \integer{{\Bbb Z}} \def \varr{{\text{var}\,}} % \def\AA{{\Cal A}} \def\BB{{\Cal B}} \def\CC{{\Cal C}} \def\DD{{\Cal D}} \def\EE{{\Cal E}} \def\FF{{\Cal F}} \def\HH{{\Cal H}} \def\II{{\Cal I}} \def\JJ{{\Cal J}} \def\KK{{\Cal K}} \def\LL{{\Cal L}} \def\MM{{\Cal M}} \def\NN{{\Cal N}} \def\OO{{\Cal O}} \def\PP{{\Cal P}} \def\QQ{{\Cal Q}} \def\RR{{\Cal R}} \def\SS{{\Cal S}} \def\TT{{\Cal T}} \def\UU{{\Cal U}} \def\VV{{\Cal V}} \def\XX{{\Cal X}} \def\ZZ{{\Cal Z}} %!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! %=============postscript======= %figure number psfile caption % (will be centered) \def\figure #1 #2 #3\cr { \bigskip \centerline{\psfig {figure=#2} } \smallskip \vbox{\eightpoint\noindent {\bf Figure #1} #3} \bigskip } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \topmatter \title On the spectra of\\ randomly perturbed expanding maps \endtitle \author V. Baladi and L.-S. Young \endauthor \address UMPA, ENS Lyon (CNRS, UMR 128), 46, All\'ee d'Italie, F-69364 Lyon Cedex, France \endaddress \email baladi\@umpa.ens-lyon.fr\endemail \address Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA \hfill\break and \hfill\break \indent Department of Mathematics, UCLA, Los Angeles, CA 90024, USA \endaddress \email lsy\@math.arizona.edu and lsy\@math.ucla.edu \endemail \date{November 1992} \enddate \subjclass 58F11 58F30; 58C25 58F03 58F15 58F19 58G32 60J10 \endsubjclass \abstract {We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving the robustness of the spectra of their Perron-Frobenius operators.} \endabstract \thanks {V. Baladi started the present work during a postdoctoral fellowship at IBM, T.J.~ Watson Center.} \endthanks \thanks {L.-S. Young is partially supported by the National Science Foundation.} \endthanks \endtopmatter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \document \head Introduction \endhead Let $f: M \to M$ be a dynamical system preserving some natural probability measure $\mu_0$ with density $\rho_0$. This paper is motivated by the following question: {\it does exponential mixing imply stochastic stability}? Roughly speaking, {\it exponential mixing} of $(f, \mu_0)$ means that, for two observables $\varphi$ and $\psi$ on $M$, the correlation between $\varphi \circ f^n$ and $\psi$ decays exponentially fast with $n$. {\it Stochastic stability} means that, if we add a small amount of random noise to $f$, obtaining at noise level $\epsilon$ a Markov process with invariant density $\rho_\epsilon$, then $\rho_\epsilon$ tends to $\rho_0$ as $\epsilon$ tends to zero. The following heuristic argument suggests an affirmative answer to this question. Consider the Perron-Frobenius operator $\LL$ associated with $f$ acting on a suitable class of functions. The exponential mixing property is equivalent to the presence of a gap in the spectrum of $\LL$ between the eigenvalue equal to unity and the ``next largest eigenvalue.'' Corresponding to the noisy situation is a noisy Perron-Frobenius operator $\LL_\epsilon$, which should not be too different from $\LL$ for small $\epsilon$. By standard perturbation arguments for linear operators, the eigenfunction corresponding to the eigenvalue $1$ for $\LL_\epsilon$ should be near that for $\LL$, proving stochastic stability. Also, since the ``second largest'' eigenvalue of $\LL$ determines the rate of decay of correlations, if there is a gap between the ``second largest'' and the ``third largest'' eigenvalue, then a similar reasoning will show that the presence of small amounts of noise should not affect significantly the rate of mixing of the system. When further gaps exist, this reasoning can be extended to some other eigenvalues of $\LL$ (the ``resonances'' of Ruelle [1986]). \smallskip One obvious way to make this heuristic argument rigorous would be to show that $\LL_\epsilon$ converges to $\LL$ in the topology of operator norms. That, unfortunately, is almost never true. In general, the relation between $\LL_\epsilon$ and $\LL$ depends on the dynamics as well as the function space in question. The purpose of this paper is to examine the nature of this perturbation for the following three models: Our first model consists of expanding maps of the circle, which we perturb by taking convolutions with a fixed kernel. The function space on which our Perron-Frobenius operators act is the space of $\CC^r$ functions. Our second model is a slight generalization of the first: we consider expanding maps of Riemannian manifolds followed by stochastic flows. Our third model consists of piecewise expanding maps of the interval, which we assume to be mixing. The perturbations are the same as those in the first model, but our test functions are only of bounded variation. All three models, when unperturbed, have the exponential mixing property. For the first two models we prove that $\LL_\epsilon$ converges to $\LL$ in a strong enough sense to guarantee the convergence of the spectrum on certain regions of the complex plane. The situation in the third model is somewhat more delicate. We have the same results provided we further restrict the domain of convergence. As explained earlier, these convergence results allow us to read off immediately properties such as stochastic stability, robustness of the rate of mixing, etc. Not all of our results are new. Stochastic stability, particularly in the sense of weak convergence of measures, has been proved for various dynamical systems. See e.g. Kifer [1988a]. Stability in the bounded variation case is first proved in Keller [1982]. More references will be given later on. This paper is organized as follows. In Section 2 we prove some simple perturbation lemmas for abstract operators. We deal with our three models in Sections 3, 4 and 5, proving some dynamical lemmas that relate $\LL_\epsilon$ to $\LL$. We then obtain our desired conclusions by appealing to the results in Section 2. We hope that this method of proof goes beyond the situations considered in the present article. In a forthcoming paper by the first named author some of the results here will be brought to greater generality. Transfer operators with more general weights will be considered, and the Fredholm determinants of the perturbed operators will be shown to converge to that of $\LL$ on certain regions of the complex plane. \medskip We express our thanks to Pierre Collet and Fran\c {co}is Ledrappier for very useful conversations. V.~Baladi acknowledges the hospitality and financial support of the U.C.L.A., the I.H.E.S., and the Niels Bohr Institute. L.-S.~Young is grateful to the Mittag-Leffler Institute for its hospitality and support. \smallskip \head 1. Background, definitions and notations \endhead Let $f: M \to M$ be a differentiable or piecewise differentiable transformation of a compact Riemannian manifold. Assume that $f$ preserves a Borel probability measure $\mu_0$ of the form $\mu_0 = \rho_0 \, dm$, where $m$ denotes Riemannian volume. Our aim in this work is to study the invariant density and rate of mixing of $(f, \mu_0)$ under small random perturbations, and we do that by studying the spectral properties of the perturbed Perron-Frobenius operators associated with $f$. The purpose of this section is to give precise definitions for all of these terms. \medskip Let $\BB$ denote the $\sigma$-algebra of Borel sets of $M$ and $\PP$ the space of Borel probability measures on $M$. Recall that a random perturbation of $f$ is a family of Markov chains $\XX^\epsilon$ (with small $\epsilon>0$) defined on the measure space $(M,\BB)$, with transition probabilities $\{P^\epsilon(x,\cdot)\}$ in $\PP$, i.e., $P\{ \XX^\epsilon_{n+1} \in E \, : \, \XX^\epsilon_n = x \} = P^\epsilon(x, E)$. We assume that the following conditions are satisfied: \roster \item The map $x \mapsto P^\epsilon(x,\cdot)$ is continuous for each $\epsilon$. \item Each $P^\epsilon(x, \cdot)$ is absolutely continuous with respect to Lebesgue measure $m$. \item For any continuous test function $g:M\to\real$ $$ \lim_{\epsilon \to 0}\, \biggl( \sup_{x \in M} |\int_M g(y) P^\epsilon(x,dy) -g(f x)|\biggr ) \, = \, 0 \, . $$ \endroster If $M$ is compact, it follows from \therosteritem{1} and \therosteritem{2} that each Markov chain $\XX^\epsilon$ admits an absolutely continuous invariant probability measure $\mu_\epsilon$, i.e., a probability measure $\mu_\epsilon = \rho_\epsilon dm$ such that $$ \mu_\epsilon (E) = \int P^\epsilon(x,E) d\mu_\epsilon(x) \, , \forall E \in \BB\, . $$ (For more details, see e.g. Kifer [1988a]. Note that the assumption that $P^\epsilon(x,\cdot)$ has a density with respect to Lebesgue is not essential for most of the results below.) \smallskip We say that $(f, \mu_0)$ is {\it stochastically stable} under the perturbation $\XX^\epsilon$ if $\mu_\epsilon$ tends to $\mu_0$ weakly as $\epsilon \to 0$. Various dynamical systems have been shown to be stochastically stable in this sense (see e.g. Kifer [1974] and the results and references in [1988a], Benedicks-Young [1992] etc.). Sometimes, one has a stronger notion of stochastic stability. If $(\FF, \| \cdot \|)$ is a Banach space of functions $\rho: M \to \real$ containing $\rho_0$ and $\rho_\epsilon$, then we say that $(f, \mu_0)$ is {\it stochastically stable in $(\FF, \| \cdot \|)$} if $\| \rho_\epsilon -\rho_0\|$ tends to zero as $\epsilon \to 0$ . (See e.g. Keller [1982] and Collet [1984] for certain interval maps, with $\FF = L^1(dm)$.) We are also going to consider the convergence of the rate of mixing. Recall that one says that $\tau_0$ is the {\it rate of decay of correlations of $(f,\mu_0)$ for functions in $(\FF, \| \cdot \|)$} if $\tau_0$ is the smallest number such that the following holds: for each $\tau > \tau_0$ and each pair $\varphi,\psi \in \FF$, there exists $C=C(\tau, \|\varphi\|,\|\psi\|)$ such that $$ | \int (\varphi \circ f^n) \cdot \psi d \mu_0 - \int \varphi d\mu_0 \int \psi d \mu_0 | \le C \tau^n \, , \;\; \forall n \ge 1 \, . $$ We are mostly interested in the case where $\tau_0 < 1$. Consider now the Markov chain $(\XX^\epsilon, \mu_\epsilon)$, and let $P^\epsilon_n(x, \cdot)$ be the $n$-step transition probability. We say that $\tau_\epsilon$ is the {\it rate of decay of correlations of $(\XX^\epsilon, \mu_\epsilon)$ for functions in $(\FF, \|\cdot \|)$} if $\tau_\epsilon$ is the smallest number such that the following holds: for each $\tau > \tau_\epsilon$ and each pair $\varphi,\psi \in \FF$, there exists $C=C(\tau, \|\varphi\|, \|\psi\|)$ such that $$ | \int (\int \varphi(y) P^\epsilon_n(x,dy)) \cdot \psi(x) d \mu_\epsilon(x)- \int \varphi d\mu_\epsilon \int \psi d \mu_\epsilon | \le C \tau^n \, , \;\; \forall n \ge 1 \, . $$ We say that the rate of mixing of $(f, \mu_0)$ in $\FF$ is {\it robust} if $\tau_\epsilon$ tends to $\tau_0$ as $\epsilon$ goes to zero. (The relation between $\tau_\epsilon$ and $\tau_0$ has been considered in e.g. Ruelle [1986], for mixing Anosov flows.) \medskip Next we define the Perron-Frobenius operator associated with $f$. For this, we fix a suitable Banach space of functions $(\FF, \| \cdot \|)$ as above, and for $\varphi \in \FF$, we define $$ \LL \varphi (x) = \sum_{f(y)=x} {\varphi(y) \over |\det Df_y|} \, . $$ Or, equivalently, if $\varphi \in \FF$ is the density of a signed measure $\mu$ on $M$, then $\LL \varphi$ is the density of $f_* \mu$ where $f_* \mu$ is the push-forward of $\mu$ by $f$, i.e., $(f_* \mu) (E) = \mu (f^{-1} E)$, for all $E \in \BB$. We assume that $\LL : \FF \to \FF$ is a well-defined bounded operator, and that $\rho_0 \in \FF$. Then $1$ is an eigenvalue of $\LL$, and our invariant density $\rho_0$ is an eigenfunction for the eigenvalue $1$. In our models, as in virtually all situations where the spectrum of the Perron-Frobenius operator is understood, the operator $\LL$ is quasi-compact, i.e., its essential spectral radius $\text{ ess sp } (\LL)$ is strictly less than its spectral radius. In particular, for every $\tau > \text{ ess sp } (\LL)$, the set $\sigma(\LL) \cap \{ z \, : \, |z| \ge \tau \}$ consists of a finite number of eigenvalues with finite dimensional eigenspaces. If we further assume that $(f,\mu_0)$ is exact~ ---~ which is the case for the models considered in this paper~---~ then it has been shown that the spectrum of $\LL$ can be written as $\sigma(\LL) = \sigma_0 \cup \{ 1\}$, where $1$ is a simple eigenvalue (i.e. it has a one-dimensional generalized eigenspace) and $|\sigma_0| := \sup \{ |z| \, : \, z \in \sigma_0 \} < 1$ (see Hofbauer--Keller [1982], Ruelle [1989]). The relationship between $\tau_0$ and $\sigma_0$ is as follows: since $$ \int (\varphi \circ f^n) \cdot \psi \, d\mu_0 = \int \varphi \cdot \LL^n( \psi \, \rho_0) \, dm \, , $$ we have $$ |\int (\varphi \circ f^n) \, \psi \, d \mu_0 - \int \varphi \, d\mu_0 \int \psi \, d \mu_0 | = | \int \varphi \bigl [ \LL^n( \psi \, \rho_0) - \bigl (\int \psi \rho_0\, dm \bigr ) \, \rho_0 \bigr ] \, dm \, | \, . $$ If $\int |\varphi| \, dm \le \text{const} \cdot \| \varphi\|$~---~and this is certainly true in our models~---~the last expression above is $$ \eqalign { \qquad&\le C \cdot \| \LL^n (\psi \rho_0) - \pi (\psi \rho_0) \| \cr &\le C' \cdot \tau^n \, , \cr } $$ where $\tau$ is any number strictly larger than $|\sigma_0|$, the constants $C$ and $C'$ depend only on $\|\varphi\|$, $\|\psi\|$ and $\tau$, and $\pi$ is the projection onto the eigenspace of $1$. Thus we have $\tau_0 = |\sigma_0|$. \smallskip If $|\sigma_0| > \text{ ess sp } (\LL)$, then $\tau_0= |\sigma_0|$ will be referred to as an {\it isolated} rate of decay. \smallskip Corresponding to the perturbation $\XX^\epsilon$ of $f$, we define the Perron-Frobenius operator $\LL_\epsilon$ as follows: if $\varphi \in \FF$ is the density of $\mu$, then $\LL_\epsilon \varphi$ is the density of $\XX^\epsilon_* \mu$ where $\XX^\epsilon_* \mu (E) = \int P^\epsilon (x, E) d\mu(x)$. Moreover, if $\rho_\epsilon \in \FF$, if $1$ is the only point of $\sigma(\LL_\epsilon)$ on the unit circle, and if it is a simple eigenvalue, then we can write $\sigma(\LL_\epsilon) =\{ 1 \} \cup \sigma_0(\LL_\epsilon)$ and the interpretation of $\tau_\epsilon$ as $|\sigma_0 (\LL_\epsilon)|$ carries over as before. In the next three sections, we will consider for each of our models the following questions: \roster \item does $\| \rho_\epsilon - \rho_0\| \to 0$? \item does $\tau_\epsilon \to \tau_0$ (assuming that $\tau_0$ is an isolated rate of decay)? \endroster If the answers to \therosteritem{1} and \therosteritem{2} are affirmative then we may also ask \roster \item[3] how does $\| \rho_\epsilon - \rho_0\|$ or $|\tau_\epsilon-\tau_0|$ scale with $\epsilon$ as $\epsilon \to 0$? \endroster \smallskip \head 2. Perturbation lemmas for abstract operators \endhead \smallskip Let $(X, \| \cdot \|)$ be a complex Banach space, and let $\{ T_\epsilon, \epsilon \ge 0 \}$ be a family of bounded linear operators on $X$. We make the following assumption about $T_0$: There exist two real numbers $0 < \kappa_1 < \kappa_0\le1$ such that the spectrum of $T_0$ decomposes as $\Sigma_0 \cup \Sigma_1$ where $$ \eqalign{ \kappa_0 &= \inf \{ |z| \, : \, z \in \Sigma_0 \} \cr \kappa_1 &= \sup \{ |z| \, : \, z \in \Sigma_1 \} \, .\cr } \tag{A.1} $$ Let $X_i$ be the eigenspace corresponding to $\Sigma_i$, and let $\pi_i : X_0 \oplus X_1 \to X_i$ be the associated projection. Let $\sigma(\cdot)$ denote the spectrum of an operator. Our first result is \proclaim{Lemma 1} Assume that there exists $\kappa < \kappa_0$ such that for each sufficiently large $n \in \integer^+$, there exists $\epsilon(n)$ such that for all $0 < \epsilon < \epsilon(n)$ $$ \| T_\epsilon^n -T_0^n \| \le \kappa^n \, . \tag{A.2} $$ Then, for each sufficiently small $\epsilon>0$, there exists a decomposition of $\sigma(T_\epsilon)$ into $$ \sigma(T_\epsilon) = \Sigma_0^\epsilon \cup \Sigma_1^\epsilon $$ such that if $$ \kappa_1^\epsilon := \sup \{ |z| \, : \, z \in \Sigma_1^\epsilon \} \text{ and } \kappa_0^\epsilon := \inf \{ |z| \, : \, z \in \Sigma_0^\epsilon \} \, , $$ then $\kappa_1^\epsilon < \kappa_0^\epsilon$. \endproclaim It will become clear later on that \thetag{A.2} agrees with the nature of our perturbations. Note that we do not assume that $T_\epsilon^n x$ converges to $T_0^n x$ as $\epsilon \to 0$ for fixed $n$ and/or $x$, nor do we assume that for fixed $\epsilon$ we know anything about $\| T_\epsilon^n - T_0^n\|$ for all large $n$. \demo{Proof of Lemma 1} Fix $\kappa'_1$, $\kappa'$ near $\kappa_1$, $\kappa$, and $\kappa_0'$, $\kappa_0''$ near $\kappa_0$ such that $$ \kappa_1 < \kappa'_1 < \kappa < \kappa' < \kappa'_0 < \kappa_0'' < \kappa_0 \, . $$ Let $N$ be large enough for all the purposes below, in particular, we require that $$ \eqalign { x \in X_0 &\Longrightarrow \| T_0^N x \| \ge (\kappa_0'')^N \| x \| \cr x \in X_1 &\Longrightarrow \| T_0^N x \| \le (\kappa_1')^N \| x \| \, . \cr } $$ Let $\epsilon < \epsilon(N)$, and let $\lambda$ satisfy $\kappa' < |\lambda| < \kappa'_0$. We will show that $\lambda \notin \sigma(T_\epsilon)$. It suffices to prove that the resolvent $R(T_\epsilon^N,\lambda^N)$ exists as a bounded operator. We write down what it must be if it exists: $$ \eqalign { R(T_\epsilon^N,\lambda^N) &= \bigl [ (\lambda^N I - T_0^N ) - (T_\epsilon^N - T_0^N) \bigr ]^{-1} \cr &= \biggl [ (\lambda^N I -T_0^N) \cdot \bigl ( I- R(T_0^N, \lambda^N) (T_\epsilon^N-T_0^N) \bigr ) \biggr ]^{-1} \cr &= \sum_{n=0}^\infty \bigl ( R(T_0^N, \lambda^N) (T_\epsilon^N-T_0^N) \bigr)^n \cdot R(T_0^N, \lambda^N) \, . } \tag 2.1 $$ Assuming $\|T_\epsilon^N -T_0^N \| < \kappa ^N$, it is enough to show $\|R(T_0^N, \lambda^N) \| < (1/\kappa)^N$. Since $R(T_0^N,\lambda^N) X_i = X_i$ for $i=0,1$, we have for $x \in X$, $\|x \|=1$ $$ \eqalign { \| R(T_0^N, \lambda^N) \| &\le \| R(T_0^N,\lambda^N) \pi_0 x\| + \| R(T_0^N, \lambda^N)\pi_1 x \| \cr &\le \| R(T_0^N, \lambda^N)|_{X_0} \| \| \pi_0\| + \| R(T_0^N, \lambda^N)|_{X_1} \| \| \pi_1\|\, . \cr } $$ so that it suffices to bound $\|R(T_0^N, \lambda^N)|_{X_i} \|$, $i=0,1$. For $x \in X_0$, we have $$ \eqalign { \| T_0^N x - \lambda^N x\| & \ge \| T_0^N x\| - | \lambda|^N \| x \| \cr &\ge \bigl ( (\kappa''_0)^N - (\kappa'_0)^N \bigr ) \| x \| \cr &\ge C \cdot (\kappa''_0)^N \| x \| \, , \cr } $$ where $C$ is a constant depending only on $\kappa_0'$ and $\kappa''_0$. This gives $$ \| R(T_0^N, \lambda^N)|_{X_0}\| \le {1 \over C (\kappa''_0)^N} \, . $$ Similarly, for $x \in X_1$, we have $$ \| T_0^N x - \lambda^N x\| \ge \bigl ((\kappa')^N - (\kappa'_1)^N \bigr ) \| x \| \, , $$ proving $$ \| R(T_0^N, \lambda^N)|_{X_1}\| \le {1 \over C (\kappa')^N} \, . $$ Hence, for large enough $N$, $$ \| R(T_0^N, \lambda^N)\| \le {\text{const} \cdot (\|\pi_0\| + \| \pi_1\|) \over (\kappa')^N} \le { 1 \over \kappa^N} \, . \tag2.2 $$ Define $$ \Sigma_0^\epsilon := \{ z \in \sigma(T_\epsilon) \, : \, |z | \ge \kappa'_0 \} \, \qquad \Sigma_1^\epsilon := \{ z \in \sigma(T_\epsilon) \, : \, |z | \le \kappa' \} \,. \qed $$ \enddemo \smallskip Note that $\kappa_1^\epsilon \le \kappa'$, which can be made arbitrarily near $\max(\kappa,\kappa_1)$ by choosing $\epsilon$ small. \smallskip Let $\pi_0^\epsilon : X_0^\epsilon \oplus X_1^\epsilon \to X_0^\epsilon$ be the projection associated with the spectral decomposition of $T_\epsilon$. For $\Gamma \subset \complex$ write $\Gamma^N := \{ z^N \, : \, z \in \Gamma \}$. We also use the notation $B_r := \{ |z|=r \}$. \proclaim{Lemma 2} If Assumptions \thetag{A.1} and \thetag{A.2} hold then $\| \pi_0 - \pi_0^\epsilon \| \to 0$ as $\epsilon \to 0$. \endproclaim \demo{Proof of Lemma 2} Note that $\pi_0$ can be regarded as the projection associated with $(T^N, (\Sigma_0)^N)$ for any $N$, and similarly for $\pi_0^\epsilon$. We will again consider $N$ large and $\epsilon < \epsilon(N)$. Let $C := B_{{\hat \kappa}^N} \cup B_{r_0^N}$ for some $\kappa' < \hat \kappa < \kappa_0'$ with $\hat \kappa < (\kappa')^2/\kappa$, and $r_0 > |\sigma(T_0)|$. Then $\Sigma_0^N$ and $(\Sigma_0^\epsilon)^N$ are contained in the annular region bounded by $C$, and we have $$ \pi_0 = {1 \over 2 i \pi} \int_C R(T_0^N,\lambda) \, d\lambda \qquad\qquad \pi_0^\epsilon = {1 \over 2 i \pi} \int_C R(T^N_\epsilon,\lambda) \, d\lambda \, . $$ We will estimate $\| \pi_0 - \pi_0^\epsilon\|$ by $$ \eqalign { \| \pi_0 - \pi_0^\epsilon\| & \le {1 \over 2 \pi} \int_C \| R(T_0^N, \lambda) - R(T^N_\epsilon, \lambda)\|\, d\lambda\cr &\le {1 \over 2 \pi} \cdot \ell(B_{{\hat \kappa}^N}) \cdot \max_{\lambda \in B_{{\hat \kappa}^N}} \| R(T_0^N, \lambda) - R(T^N_\epsilon, \lambda)\| \cr &\quad + \text{ the corresponding term for $B_{r_0^N}$} \cr &=: (1) + (2) \, . \cr }\tag 2.3 $$ Using \thetag{2.1} we have $$ \| R(T_0^N, \lambda) - R(T^N_\epsilon, \lambda)\| \le \sum_{n=1}^\infty \| R(T_0^N, \lambda)\|^{n+1} \cdot \| T_\epsilon^N - T_0^N\|^n \, . $$ Since $\ell(B_{{\hat\kappa}^N}) = 2 \pi {\hat\kappa}^N$, and $\| R(T_0^N, \lambda) \| \le \text{const} / (\kappa')^N$ for $\lambda \in B_{\hat \kappa^N}$ (by \thetag{2.2}), we obtain $$ \eqalign { (1) &\le {\hat\kappa}^N \cdot \sum_{n=1}^\infty \biggl ( {\text{const} \over {\kappa'}^N} \biggr )^{n+1} (\kappa^N)^n \cr &\le \text{const} \cdot {\hat\kappa}^N \cdot {\kappa^N \over ({\kappa'}^N)^2} \to 0 \text{ as } N \to \infty \, . } $$ For $(2)$, we use $\ell(B_{{r_0}^N}) = 2 \pi {r_0}^N$, to get $$ (2) \le \text{const} \cdot r_0^N \cdot {\kappa^N \over r_0^{2N}} \to 0 \text{ as } N \to \infty\, .\, \qed $$ \enddemo For $n \ge 1$ define $$ C_n(\epsilon) := \sup \Sb x \in X_0 \\ x \ne 0 \endSb { \|T^n_\epsilon x - T^n_0 x \| \over \| x\| } \, . $$ (By \thetag{A.1}, $C_n(\epsilon) < \kappa^n$ for large enough $n$ and small enough $\epsilon$.) \proclaim{Lemma 3} Assume that \thetag{A.1}-\thetag{A.2} hold, that $\| T_\epsilon\|$ is uniformly bounded, and that $$ \text{dim } X_0 < \infty \, . \tag{A.3} $$ Let $d$ denote the maximum algebraic multiplicity of $z \in \sigma(T_0|_{X_0})$ and let $\kappa'$ and $\kappa'_0< \kappa_0$ be given from Lemma~1. Then for fixed large $N$ and $\epsilon < \epsilon(N)$: \roster \item $\text{Hausdorff-distance} (\sigma(T_0|_{X_0}), \sigma(T_\epsilon|_{X_0^\epsilon})) \le \text{const} \cdot ( C_1(\epsilon) + {C_N(\epsilon)\over {\kappa'_0}^N})^{1/d}$. \item If $\hat x_0 \in X_0$ is an eigenvector for $T_0$ with $T_0 \hat x_0 = \nu_0 \hat x_0$, then $T_\epsilon$ has an eigenvector $\hat x_0^\epsilon \in X_0^\epsilon$ with eigenvalue $\nu_0^\epsilon$ which is $\text{const} \cdot ( C_1(\epsilon) + {C_N(\epsilon)\over {\kappa'_0}^N})^{1/d}$-near $\nu_0$ such that $$ \| \hat x_0^\epsilon - \hat x_0 \| \le \text{const} \cdot ( C_1(\epsilon) + {C_N(\epsilon)\over {\kappa'_0}^N})^{1/d} \, . $$ \endroster \endproclaim The assumption that $\|T_\epsilon\|$ is uniformly bounded is not essential since for some large iterate $\| T_\epsilon^N\| \le \|T_0^N\| +\kappa^N$ for all small enough $\epsilon$. \smallskip \demo{Proof of Lemma 3} First we show that $X_0^\epsilon = \text{graph} (S_\epsilon)$ for some linear $S_\epsilon : X_0 \to X_1$ with $\| S_\epsilon \| \to 0$ as $\epsilon \to 0$. To see this, consider $\epsilon$ small and let $x \in X_0^\epsilon$. Since $\| x - \pi_0 x \| \le \| \pi_0^\epsilon - \pi_0 \| \| x\|$, it follows that if $x = (x_0, x_1) \in X_0 \oplus X_1$, then $\| x_1\| \ll \| x_0 \|$. This inequality implies in particular that if $x$, $x' \in X_0^\epsilon$ and $\pi_0 x = \pi_0 x'$ then $x = x'$. Next, we estimate $\| S_\epsilon\|$. We know by \thetag{A.3} that there exists $x_0 \in X_0$, $\| x_0\|=1$, such that $$ \| S_\epsilon \| \le {\| \pi_1 T_\epsilon^N (x_0, S_\epsilon x_0) \| \over \| \pi_0 T_\epsilon^N(x_0, S_\epsilon x_0) \| } \, . $$ This is $$ \phantom{\| S_\epsilon \| } \le { \| \pi_1 \| \biggl ( \bigl ( (\kappa'_1)^N + \kappa^N \bigr ) \| S_\epsilon \| + C_N(\epsilon) \biggr ) \over (\kappa'_0)^N - \| \pi_0 \| (1 + \| S_\epsilon \| ) \cdot \kappa^N } \, , \tag{2.4} $$ from which we see that $$ \| S_\epsilon \| \le \text{const} { C_N(\epsilon) \over (\kappa'_0)^N } \, . $$ \smallskip Define $\hat T_\epsilon : X_0 \to X_0$ by $$ \hat T_\epsilon(x) = \pi_0 \circ T_\epsilon(x, S_\epsilon x) \, . $$ Then for $x \in X_0$ with $\| x \|=1$, we have $$ \eqalign { \| \hat T_\epsilon x - T_0 x \| &\le \|\pi_0\| \cdot (\|T_\epsilon x - T_0 x\| + \| T_\epsilon S_\epsilon x \| ) \cr &\le \text{const} \cdot ( C_1(\epsilon)+ \| T_\epsilon\| \cdot {C_N(\epsilon) \over {\kappa'_0}^N}) \, . } $$ There is a similar bound for $\| \pi_1 \circ T_\epsilon(x,S_\epsilon x) - \pi_1 T_0 x \|$ with $x \in X_0$. The assertions of Lemma 3 follow immediately. (See e.g. Wilkinson [1965].) \qed \enddemo \smallskip \head 3. The simplest model: \\ expanding maps of the circle and perturbations by convolutions \endhead \subhead A. The unperturbed model \endsubhead \smallskip Assume first that our manifold $M$ is equal to the circle $S^1$. Let $f$ be a $\CC^r$ transformation of $S^1$ ($2 \le r < \infty$) which is expanding, i.e., $|f'| \ge \lambda >1 $. The {\it expanding constant} of $f$ is the largest $\lambda$ such that this inequality holds. This implies the existence of a unique absolutely continuous invariant probability measure $\mu_0$ with respect to which $f$ is mixing (in fact, exact). We set $\FF = \CC^{r-1}(S^1)$ and let $\| \cdot \|$ be the usual $\CC^{r-1}$-norm. Let $\LL: \FF \to \FF$ be the Perron--Frobenius operator associated with $f$: $$ \LL\varphi(x) = \sum_{f (y) = x} {\varphi(y) \over |f'(y)|} \, . $$ It is proved in Ruelle [1989] (see also Collet--Isola [1991]) that $\LL$ is quasi-compact with essential spectral radius bounded above by $(1/\lambda)^{r-1}$. \medskip We remark that if the map $f$ is $\CC^\infty$ or $\CC^\omega$, we can let $\LL$ act on the Fr\'echet space $\CC^\infty(S^1)$ of $\CC^\infty$ functions, respectively the Banach space $\CC^\omega(S^1)$ of real analytic functions endowed with the supremum norm. Using the fact (Ruelle [1989]) that, for a $\CC^r$ map, the eigenfunctions of $\LL$ acting on $\CC^{r'}$ for $1 \le r' < r-1$ are all elements of $\CC^{r-1}(S^1)$, it makes sense to speak of the eigenvalues of $\LL$ when acting on $\CC^\infty(S^1)$, even though $\CC^\infty(S^1)$ is not a Banach space. In particular, one can view $\LL:\CC^\infty(S^1)\to \CC^\infty(S^1)$ as a ``compact'' operator. If $r=\omega$, the operator $\LL$ is (truly) compact, and much is known about it (Ruelle [1976], Mayer [1976], etc.). We will not discuss further the cases $r=\infty, \omega$, but our results clearly hold there too. \medskip We remark also that $\tau_0=|\sigma_0|$ is not always an isolated rate of decay. Consider for instance the map $z \to z^2$ on $S^1$ and its the transfer operator acting on real analytic functions. By following the computation in Ruelle [1986], one checks that the relevant Fredholm determinant is equal to $(1-z)$, so that the only eigenvalue is $1$. This implies (Ruelle [1976,1989,1990]) that the transfer operator acting on $\CC^{r}(S^1)$, with $1 \le r \le \infty$ has no eigenvalue besides $1$ whose modulus is bigger than the essential spectral radius. The other ``algebraic'' maps $z \mapsto z^k$, for integers $k \ge 3$, have the same property. However, as pointed out to us by Mark Pollicott, the above examples do not seem to be generic: a necessary condition for the lack of nontrivial eigenvalues in the spectrum of the operator acting on analytic functions is the fact that the trace of the Fredholm operator is equal to $1$. By considering analytic perturbations of the algebraic examples, one can arrange that the value of this trace changes. For example, the projection on the circle of the periodic map $x \mapsto 2x (\text{mod}\, 1) + \delta \sin 2\pi x$ only has one fixed point (if $\delta> 0$ is not too large), and the trace of its Perron-Frobenius operator can easily be computed to be $1/(1-\delta) > 1$, so that there is at least one eigenvalue besides $1$ whose real part is strictly positive (Pollicott [1991]). \subhead B. Type of perturbation: convolutions \endsubhead \smallskip For $\epsilon > 0$, let $\theta_\epsilon : \real \to \real$ be a function in $L^1(dm)$ satisfying $$ \theta_\epsilon \ge 0 \, , \; \text{supp} \, \theta_\epsilon \subset [-\epsilon,\epsilon] \, , \; \text{and} \int \theta_\epsilon = 1 \, . $$ Consider the random perturbation $\XX^\epsilon$ where the transition probabilities $P^\epsilon(x,dy)$ have densities $\theta_\epsilon (y-fx)$. (Note that the density depends only on the difference $y-fx$.) Equivalently, using Fubini's Theorem, one can describe this process as given by $f$ followed by a random translation by $\omega$, where $\omega$ is distributed according to $\theta_\epsilon$. We call such a perturbation a {\it random perturbation by convolution} (see Kifer [1988a, Chapter IV]). \medskip The perturbed Perron--Frobenius operator $\LL_\epsilon :\CC^{r-1}(S^1) \to \CC^{r-1}(S^1)$ can be written as follows: for $\varphi \in \CC^{r-1}(S^1)$, $$ \align (\LL_\epsilon \varphi) (x) &= \int (\LL \varphi) (x-\omega) \theta_\epsilon (\omega)\, d\omega \cr &=\int \varphi(y) \theta_\epsilon (x - fy) dm(y) \, .\cr \endalign $$ Analogous operators have been used by Keller [1982, \S 5] and Collet [1984] among others. The operator $\LL_\epsilon$ is clearly linear and bounded on $\CC^{r-1}(S^1)$. Also, it is quasi-compact and the density $\rho_\epsilon$ is in $\CC^{r-1}$ (Ruelle [1990]). If we had made the additional asssumption that $\theta_\epsilon$ is $\CC^{r-1}$, then $\LL_\epsilon$ would be a compact operator on $\CC^{r-1}(S^1)$. This follows from the fact that a kernel operator $$ \varphi (x) \to \int_{S^1} K(x,y) \varphi(y)\, dm(y) \, , \; \, \varphi \in \CC^0(S^1) \, , $$ with $\CC^0$ kernel $K(\cdot,\cdot)$ is compact (see e.g. Yosida [1980, p. 277]). \subhead C. Statement of our results \endsubhead \smallskip We now state our main results, which give partial answers to the questions posed in Section 1 for this simplest model: \proclaim {Theorem 1} Let $f: S^1\to S^1$ be a $\CC^r$ expanding map ($r \ge 2$) of the circle as defined in Section 3.A, with expanding constant $\lambda$, and let $\mu_0= \rho_0 \, dm$ be its unique absolutely continuous invariant probability measure. Let $\XX^\epsilon$ be a small random perturbation of $f$ of the type described in Section 3.B, with invariant measure $\mu_\epsilon = \rho_\epsilon dm$. Then: \roster \item The dynamical system $(f,\mu_0)$ is stochastically stable under $\XX^\epsilon$ in the space of $\CC^{r-1}$ functions, i.e., $\|\rho_\epsilon - \rho_0\|_{r-1}$ tends to $0$ as $\epsilon \to 0$. Moreover, we have $\| \rho_\epsilon -\rho_0\|_{r-2} = O(\epsilon)$. \endroster Let $\tau_0$ and $\tau_\epsilon$ be the rates of decay of correlations for $f$ and $\XX^\epsilon$ respectively, in the space of $\CC^{r-1}$ functions. \roster \item [2] If $\tau_0 > \lambda^{-(r-1)}$, then the rate of mixing is robust, i.e., $\tau_\epsilon \to \tau_0$ as $\epsilon \to 0$. Furthermore, if $\tau_0 > \lambda^{-(r-2)}$ then $|\tau_\epsilon -\tau_0| = O(\epsilon^{1/d})$ for some integer $d\ge 1$. \endroster We show in fact that \roster \item[3] For each $\delta > 0$, the spectrum of $\LL_\epsilon$ restricted to $\{ |z| > \lambda^{-(r-1)} + \delta\}$, converges to that of $\LL$ (restricted to the same domain) as $\epsilon \to 0$. \endroster \endproclaim The proofs below yield the same results for small {\it deterministic} perturbations by translations (i.e., maps $f^\epsilon = f + t$ with $|t|\le \epsilon$), as well as for perturbations of $\CC^r$ expanding transformations of higher-dimensional tori. \medskip \subhead D. Dynamical lemmas \endsubhead \smallskip In this section we prove the dynamical lemmas which will allow us to reduce Theorem~1 to an abstract statement about linear operators acting on Banach spaces (see Section~2). The setting and notations are as in Sections 3.A and 3.B. \proclaim{Lemma 4} \roster \item For a fixed $n \in \integer^+$ and $\varphi \in \CC^{r-1}$ $$ \| \LL_\epsilon^n \varphi - \LL^n \varphi\| \to 0 \; \; \text{as } \epsilon \to 0 \, . $$ \item For a fixed $n \in \integer^+$ and $\varphi \in \CC^{r-1}$, we have in the $\CC^{r-2}$ norm $\| \cdot \|_{r-2}$ $$ \| \LL_\epsilon^n \varphi - \LL^n \varphi\|_{r-2} = O(\epsilon)\, , \; \; \epsilon \to 0 \, . $$ \endroster \endproclaim \demo{Proof of Lemma 4} It suffices to show the lemma for $n=1$, the inductive step follows from the triangle inequality $$ \eqalign { \|\LL_\epsilon^n \varphi - \LL^n \varphi\| &= \| \LL_\epsilon (\LL_\epsilon^{n-1} \varphi) - \LL (\LL^{n-1} \varphi)\| \cr &\le \| \LL_\epsilon (\LL_\epsilon^{n-1} \varphi - \LL^{n-1} \varphi)\| + \| \LL_\epsilon (\LL^{n-1} \varphi) - \LL (\LL^{n-1} \varphi)\| \, . \cr } $$ (The induction hypothesis need only be applied to $\varphi$ and $\LL^{n-1} \varphi$.) \roster \item Since $\LL_\epsilon \varphi = \theta_\epsilon * \LL \varphi$, each derivative satisfies $D^k (\LL_\epsilon \varphi) = \theta_\epsilon * D^k(\LL \varphi)$. It hence suffices to consider $\CC^0$-norms. But if $\psi$ is continuous the convolution $\theta_\epsilon * \psi$ converges uniformly to $\psi$. \item To show the claimed asymptotic scaling in the $\CC^{r-2}$ norm, it again suffices to consider the case $r=2$. Observe that for any $\psi \in \CC^1$ the Mean Value Theorem implies $$ \eqalign { |\theta_\epsilon * \psi (x) - \psi(x)| &\le \int \theta_\epsilon(t)| (\psi(x-t)-\psi(x))| \, dt\cr &\le\sup_\xi |\psi'(\xi)| \cdot \int \theta_\epsilon(t) \cdot t \, dt\cr &\le \sup_\xi |\psi'(\xi)| \cdot 2 \epsilon \, .\qed\cr } $$ \endroster \enddemo \smallskip We want to emphasize that in general $\LL_\epsilon$ does {\it not} converge to $\LL$ in the operator topology when $\epsilon \to 0$. (For example, if $\theta$ is $\CC^{r-1}$, the operators $\LL_\epsilon$ are all compact and convergence in norm would imply that $\LL$ is compact too --- but this is well-known to be false: see the explicit construction of essential spectral values in Collet--Isola [1991].) \medskip The key lemma follows: \proclaim{Lemma 5} Let $\Lambda > \lambda^{-(r-1)}$ be given. Then there exists $N_0 \in \integer^+$ such that for each $n \ge N_0$, there exists $\epsilon(n) > 0$ such that for each $\epsilon < \epsilon(n)$, one has $$ \| \LL_\epsilon ^n - \LL^n \| < \Lambda^n \, . $$ \endproclaim \smallskip \demo {Proof of Lemma 5} We use the following notations: $C$ represents a constant independent of $n$ and $\epsilon$; $c_{n,\epsilon}$ represents a constant depending only on $n$ and $\epsilon$ (and not on test functions), and tending to zero as $\epsilon\to 0$ for each fixed $n$. We also write $g$ for $1/|f'|$. Recall that $$ \eqalign{ (\LL^n \varphi) (x) &= \sum_{y : f^n y = x} \varphi(y) (g(y) \cdot g(fy) \cdots g(f^{n-1} y) \cr &=\sum_{y : f^n y = x} (\LL^n \varphi_y) \, , \cr } $$ where the second equality defines $(\LL^n \varphi_y)$. Writing, for $\vec t =(t_1, \ldots, t_n)$, $$ f^n_{\vec t}(z) = f(\ldots f(f(z)+t_1)+t_2) \ldots)+t_n \, , $$ we have $$ \eqalign { (\LL^n_\epsilon \varphi) (x) &=\int \cdots \int dt_1 \ldots dt_n \, \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n) \sum_{y_{\vec t} : f^n_{\vec t} (y_{\vec t}) = x}\varphi(y_{\vec t}) g(y_{\vec t}) \cdots g(f^{n-1}_{\vec t} y_{\vec t}) \cr &= \int \cdots \int dt_1 \ldots dt_n \, \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n) \sum_{y_{\vec t} : f^n_{\vec t}(y_{\vec t}) = x} (\LL_{\vec t}^n \varphi)_{y_{\vec t}} \cr &= \int \cdots \int dt_1 \ldots dt_n \, \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n)\, (\LL_{\vec t}^n \varphi) (x) \, ,\cr } $$ where the last two equalities define $(\LL_{\vec t}^n \varphi)$ and $(\LL_{\vec t}^n \varphi)_{y_{\vec t}}$. We have used the fact that all orbits are {\it strongly shadowable}: that is, if $\epsilon$ is small enough, then for a fixed $x$ and a fixed $n$-tuple $(t_1, \ldots, t_n)$ with $|t_i| \le \epsilon$, there is a natural bijection between the set $\{y \, :\, f^n (y)=x\}$ and the set $\{ y_{\vec t} \, : \, f^n_{\vec t}(y_{\vec t}) = x \}$. Moreover, for each pair $(y,y_{\vec t})$ corresponding to a choice of an inverse branch of $f^n$ at $x$ we have $$ g(y) \cdot g(fy) \cdots g(f^{n-1} y) =g(y_{\vec t}) \cdot g(f_{\vec t} y_{\vec t}) \cdots g( f^{n-1}_{\vec t} y_{\vec t}) \pm c_{n,\epsilon} \, . \tag3.1 $$ We first show the lemma in the case $r=2$. Let us compare $\LL$ and $\LL_\epsilon$ in the $\CC^0$-norm, noting $|\varphi|=\sup |\varphi|$ and $|\varphi'|=\sup |\varphi'|$. $$ \eqalign { (\LL_{\vec t} ^n \varphi)_{y_{\vec t}} &= ( \varphi(y) \pm c_{n,\epsilon} | \varphi'|) ( \prod_{j=0}^{n-1} g(f^j y) \pm c_{n,\epsilon}) \cr &= (\LL^n \varphi)_y \pm c_{n,\epsilon} ( |\varphi| + |\varphi'|) \, .\cr } \tag3.2 $$ Hence, summing over inverse branches, and integrating over the $t_i$, $$ (\LL_\epsilon^n \varphi) (x) = (\LL^n \varphi) (x) \pm c_{n,\epsilon} \| \varphi\|_1 \, . \tag3.3 $$ We now consider first derivatives, using the Leibnitz Theorem and decomposing $$ {d \over dx }(\LL_{\vec t}^n \varphi)_{y_{\vec t}} $$ into a first part $A$ which is a sum of terms where some $g$ factor is differentiated and a second part $B$ where $\varphi$ is differentiated. For the first part we have $$ \eqalign { A&= \sum_{j=0}^{n-1} \varphi(y_{\vec t}) g( y_{\vec t}) \cdots [g'(f^j_{\vec t} y_{\vec t}) g(f^{j}_{\vec t} y_{\vec t}) \ldots g(y_{\vec t})] g(f^{j+1}_{\vec t} y_{\vec t}) \ldots g(f^{n-1}_{\vec t}y_{\vec t}) \cr &= \sum_j (\varphi(y) \pm c_{n,\epsilon} | \varphi'|) \bigl (g(y) \cdots [g'(f^j(y)) \cdots ] \cdots g(f^{n-1} y) \pm c_{n,\epsilon}\bigr ) \cr &= (\text{ the corresponding part for } {d \over dx} (\LL^n \varphi)_y \;) \pm c_{n,\epsilon} (|\varphi| + |\varphi'|) \, . } \tag3.4 $$ For the second part, we get $$ \eqalign {B&=\varphi'(y_{\vec t}) \cdot \prod_{j=0}^{n-1} g(f^j_{\vec t} y_{\vec t}) \cdot \prod_{j=0}^{n-1} g(f^j_{\vec t} y_{\vec t}) \cr &= (\varphi'(y) \pm 2 | \varphi'|) \cdot (\prod_{j=0}^{n-1} g(f^j y) \pm c_{n,\epsilon}) \cdot (\prod_{j=0}^{n-1} g(f^j y) \pm c_{n,\epsilon} ) \cr &= \varphi'(y) \bigl (\prod_{j=0}^{n-1} g(f^j y)\bigr )^2 \pm c_{n,\epsilon} | \varphi'| \pm 2 | \varphi' | \lambda^{-n} \prod_{j=0}^{n-1} g(f^j y) \, . \cr } \tag3.5 $$ Summing over inverse branches, and integrating over the $t_i$, we obtain $$ (\LL^n_\epsilon \varphi)' = (\LL^n \varphi)' \pm c_{n,\epsilon} \| \varphi\|_1 \pm 2 \| \varphi \|_1 \lambda^{-n} \sum_{y:f^n(y)=x} \prod g(f^j(y)) \, . \tag3.6 $$ Since the sum in the last term of the right-hand-side is equal to $\LL^n(1)(x)$, we know that it is uniformly bounded since $\LL^n(1)$ converges. \smallskip For arbitrary differentiability $r$, note that for $k \le r-2$, the terms of the $k$th derivative $(\LL^n\varphi)_{y}^{(k)}$ involve only the $\ell$th derivative of $\varphi$ for $\ell \le k$ so that $$ | (\LL_\epsilon^n \varphi)^{(k)} - (\LL^n \varphi)^{(k)} | \le c_{n,\epsilon} \| \varphi\|_{k+1} \le c_{n,\epsilon} \| \varphi\|_{r-1} \, . $$ The only potentially troublesome term is part $B$ of $(\LL_\epsilon^n \varphi)^{(r-1)} (x)$, i.e., $$ \int \ldots \int dt_1 \ldots dt_n \, \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n) \sum_{y_{\vec t}} \varphi^{(r-1)} (y_{\vec t}) \bigl ( \prod_j g(f^j_{\vec t} y_{\vec t}) \bigr )^{r} \, , $$ but the same argument as above yields an additional error term of the type $$ c_{n,\epsilon} \| \varphi\|_{r-1} + C \cdot \lambda^{-n(r-1)} \| \varphi\|_{r-1} \, . \quad (3.7) \, \qed $$ \enddemo \smallskip In fact, we have not used the expanding condition as stated but only a slightly weaker condition: $$ \exists \lambda > 1 \text{ such that } \lim_{n \to \infty} \bigl( \inf_x |{f^n}'(x)|^{1/n}\bigr ) > \lambda \, . $$ \subhead E. Proof of Theorem 1 \endsubhead \smallskip Unless otherwise stated we will use the results in Section 2 with $X$ the space of $\CC^{r-1}$ functions on $S^1$, $\| \cdot\|$ the $\CC^{r-1}$ norm, $T_0 =\LL$ and $T_\epsilon=\LL_\epsilon$. To prove \therosteritem{1}, we let $\Sigma_0=\{1\}$. Lemma 5 together with the fact that $(f,\mu_0)$ is exact tell us that conditions \thetag{A.1} to \thetag{A.3} in Section 2 are met. We also know that $\| \LL_\epsilon \|$ is uniformly bounded, that $1$ is always an eigenvalue of $\LL_\epsilon$ and $\rho_\epsilon$ is an eigenfunction for $1$. We conclude from Lemma 1 that $X_0^\epsilon$ must be the linear span of $\rho_\epsilon$. Lemma 3 then tells us that for any $\kappa_0' < 1$, $\|\rho_\epsilon -\rho_0\| = O( C_1(\epsilon) + {C_N(\epsilon)\over {\kappa'_0}^N})^{1/d})$ which tends to zero as $\epsilon \to 0$ by Lemma 4 \therosteritem{1}, proving stochastic stability in $(\CC^{r-1}(S^1), \| \cdot \|)$. Since $C_N(\epsilon) := \| \LL^N_\epsilon \rho_0 - \rho_0\|$, the speed with which $C_N(\epsilon)$ tends to $0$ depends on the modulus of continuity of $D^{r-1} \rho_0$. In particular, if we rewrite everything with $X=\CC^{r-2}(S^1)$ and $\| \cdot \|$ the $\CC^{r-2}$ norm, then $D^{r-2}\rho_0$ is Lipschitz and we have by Lemma 4 \therosteritem{2} $C_N(\epsilon)= O(\epsilon)$. This completes the proof of \therosteritem{1}. To prove \therosteritem{2}, we let $\Sigma_0= \sigma(\LL) \cap \{ |z| \ge \tau_0\}$. Note that conditions \thetag{A.1} and \thetag{A.2} in Section 2 are guaranteed by our assumption that $\tau_0 > \lambda^{-(r-1)} \ge \text{ess sp} (\LL)$. Since $\sigma(\LL_\epsilon) \subset (\sigma(\LL_\epsilon|_{X_0^\epsilon}) \cup \sigma(\LL_\epsilon |_{X_1^\epsilon}))$, we know that $\tau_\epsilon =\sup \{ |z| : z \in \sigma(\LL_\epsilon|_{X_0^\epsilon}), z \ne 1 \}$. Lemma 3 then tells us that for any $\tau_0' < \tau_0$, $|\tau_0 - \tau_\epsilon| = O(( C_1(\epsilon) + {C_N(\epsilon)\over {\tau'_0}^N})^{1/d})$, proving the robustness of $\tau_0$. To see how $|\tau_\epsilon -\tau_0|$ scales with $\epsilon$, we let $\LL$ act on $(\CC^{r-2} (S^1), \| \cdot \|_{r-2})$ instead of $(\CC^{r-1}(S^1), \| \cdot \|_{r-1})$. Since the eigenfunctions of $\LL$ are always $\CC^{r-1}$, the rates of decay of correlations are the same in both cases provided that $\tau_0 > \lambda^{-(r-2)}$ (note that this implies in particular $r > 2$). So even as we change the space on which $\LL$ acts, the definition of $\Sigma_0$ remains unchanged. In fact, $X_0$ stays the same (Ruelle [1989]). In the definition of $C_N(\epsilon)$, we are now dealing with $\CC^{r-2}$ norms for functions in $X_0$, a finite dimensional subspace of $\CC^{r-1}(S^1)$. By Lemma 4 \therosteritem{2}, we have $C_N(\epsilon) = O(\epsilon)$. Hence $|\tau_\epsilon - \tau_0| =O(\epsilon^{1/d})$. To prove \therosteritem{3}, let $\Sigma_0=\sigma(\LL) \cap \{|z| \ge \lambda^{-(r-1)} +\delta\}$.\qed \smallskip \head 4. Expanding maps of manifolds followed by stochastic flows \endhead This is a generalization of Section 3. \subhead A. The unperturbed model \endsubhead \smallskip Here, $M$ is a $\CC^\infty$ compact, connected Riemannian manifold without boundary, and $f : M \to M$ is a $\CC^r$ map for some $2 \le r < \infty$. We assume that $f$ is {\it expanding}, i.e., there exists $\lambda > 1$ such that for all $x$ in $M$ and all $v$ in $T_x M$, we have $| Df_x v| \ge \lambda |v|$. The largest such $\lambda$ is called the {\it expanding constant} of $f$. It is well-known that an expanding map $f$ admits a unique absolutely continuous invariant probability measure $\mu_0 = \rho_0 dm$ with respect to which $f$ is exact (see e.g. Ma\~n\'e [1987]). Let $\FF = \{ \varphi : M \to \real \, : \, \varphi \text{ is } \CC^{r-1} \}$. For $\varphi \in \FF$, we define $\| \varphi \|$ to be the $\CC^{r-1}$-norm of $\varphi$, defined using a set of charts that will remain fixed throughout. The Perron-Frobenius operator $\LL : \FF \to \FF$ is defined as usual. Ruelle's results stated in the last section are in fact proved in this more general setting. In particular, we have the inequality $$ \text{ess sp} (\LL) \le \lambda^{-(r-1)} \, . $$ \subhead B. Type of perturbation: time-$\epsilon$-maps of stochastic flows \endsubhead \smallskip Let $X_0, X_1, \ldots, X_m$ be $\CC^\infty$ vector fields on $M$, and consider the stochastic differential equation of Stratonovich type $$ d \xi_t = X_0 \, dt + \sum_{i=1}^m X_i \circ d \beta^i_t \, , \tag4.1 $$ where $\{ \beta^i_t\}$ is the standard $m$-dimensional Brownian motion. We define $\XX^\epsilon$, our $\epsilon$-perturbation of $f$, to be $\xi_\epsilon \circ f$, i.e., $\XX^\epsilon$ is the Markov chain whose transition probabilities are given by $$ P^\epsilon(x,E) = \text{Prob } \{ (\xi_\epsilon \circ f) (x) \in E \} \, . $$ If the vector fields $X_0, \ldots X_m$ span the tangent space of $M$, then condition \therosteritem{2} from Section 1 is satisfied. As in the last section, we wish to view $\XX^\epsilon$ as the composition of random maps. To do that we realize the solution of \thetag{4.1} as a stochastic flow $\{ \xi_t\}_{t \ge 0}$, i.e., we realize the solution of \thetag{4.1} as a $\text{Diff }^\infty(M)$-valued stochastic process $\{ \xi_t\}$ satisfying \roster \item"(i)" $\xi_0 = \text {Id}$, the identity map, \item "(ii)" for $t_0 < t_1 < \ldots < t_n$, the increments $\xi_{t_i}\circ \xi_{t_{i-1}}^{-1}$ are independent, \item"(iii)" for $s < t$, the composition $\xi_t \circ \xi_s^{-1}$ depends only on $t-s$, \item"(iv)" with probability $1$, the stochastic flow $\xi_t$ has continuous sample paths. \endroster (See, e.g. Kunita [1990] for more information.) Let $\nu_\epsilon$ denote the distribution of $\xi_\epsilon$ on $\text{Diff }^\infty(M)$. Then $\XX^\epsilon$ is equivalent to the random map $$ \cdots \circ (\xi_\epsilon(\omega_2) \circ f) \circ (\xi_\epsilon(\omega_1) \circ f) \, , $$ where $\xi_\epsilon(\omega_1)$, $\xi_\epsilon(\omega_2)$, ... are i.i.d. with law $\nu_\epsilon$. Using this representation of $\XX^\epsilon$, we can write the perturbed Perron-Frobenius operator $\LL_\epsilon : \CC^{r-1}(M) \to \CC^{r-1}(M)$ as follows. Let $f_\omega = \xi_\epsilon(\omega) \circ f$, then $$ (\LL_\epsilon \varphi) (x) = \int \nu_\epsilon(d \omega) (\LL_\omega \varphi) (x) \, , $$ where $$ (\LL_\omega \varphi) (x) = \sum _{y : f_\omega y = x} {\varphi(y) \over |\det Df_\omega(y)|} \, . $$ In fact, $\LL_\epsilon$ is still in the framework studied by Ruelle [1990] and in particular is quasicompact. Again, $\LL_\epsilon$ has $1$ as an eigenvalue, with eigenfuction $\rho_\epsilon\in \CC^{r-1}$ equal to the density of the invariant measure for $\XX^\epsilon$. \bigskip In the remainder of this subsection we summarize a few technical estimates about the $\CC^{r}$-norms of $\xi_\epsilon$ that will be needed later on. For $\xi \in \text{Diff }^{r} (M)$, we define the $\CC^{r}$-norm $\|\xi\|_r$ to be $\| \xi \|_r= \sum_{i=0}^{r} |D^i \xi|$, where $|D^i \xi|$ is computed using a fixed system of charts, and let $\|| \xi |\| := \max ( \| \xi \|_r, \|\xi^{-1}\|_r)$. We assume that $\|| \text{Id} |\|=1$. For $\delta > 0$, we define the sets $$ \eqalign { \UU_\delta & := \{ \xi \in \text{Diff }^{r}(M) \, : \, \|| \xi |\| < 1 + \delta \} \cr \UU_\delta^n &:= \{ \xi = \eta_n \circ \cdots \circ \eta_1 \, : \, \eta_i \in \UU_\delta\, , \forall i \} \, ,\cr } $$ and the random variable $\tau_n(\delta):= \inf \{ s \, : \, \xi_s \notin \UU^n _\delta\}$. It is proved in Baxendale [1984] and Kifer [1988b] that for all $\epsilon > 0$ $$ P\{ \tau_n (\delta) \le \epsilon \} \le ( P \{ \tau_1(\delta) \le \epsilon \} )^n \, . $$ Also, using a formula in Franks [1979, Lemma 3.2], we obtain inductively that for all $\xi$ in $\UU_\delta^n$, $$ \|| \xi |\| \le C^{n-1} (1+\delta) \biggl [ (1+\delta)^{r} +1 \biggr ] ^{n-1} , $$ where the constant $C$ only depends on $r$. From these estimates, we easily derive the following sublemmas: \proclaim{Sublemma 1} (Baxendale [1984], Kifer [1988b]). Fix $k > 0$. Then for all sufficiently small $\epsilon > 0$, the expectation $$ E (\||\xi_\epsilon |\|^k) < \infty \, . $$ \endproclaim \demo{Proof of Sublemma 1} Fix an arbitrary $\delta > 0$ and choose $\epsilon$ such that $P\{ \tau_1(\delta) < \epsilon \}$ is sufficiently small. Let $\tau_0 =0$, and define $A_n := \{ \tau_{n-1}(\delta) \le \epsilon < \tau_n (\delta)\}$. Then $$ \eqalign { E\||\xi_\epsilon |\|^k &\le \sum_{n=1}^\infty ( \sup \{ \|| \xi |\| \, : \, \xi \in \UU_\delta^n\} )^k \cdot P(A_n) \cr &\le \sum_{n=1}^\infty \biggl [ C^{n-1} (1+\delta) \bigl ( (1+\delta)^{r}+1 \bigr )^{n-1} \biggr ] ^k \cdot \biggl ( P \{ \tau_1(\delta) < \epsilon \} \biggr )^{n-1} \cr & < \infty \, . \, \qed\cr } $$ \enddemo The proof of Sublemma 1 also gives the uniform integrability of $\| | \xi_\epsilon |\|^k$ as $\epsilon$ varies. We state that as Sublemma 2. \proclaim{Sublemma 2} Fix $k > 0$ and assume $\epsilon$ is small. Then given $\alpha > 0$, there exists $\beta > 0$ (independent of $\epsilon$) such that for all $A\subset \text{Diff }^r(M)$ with $\nu_\epsilon(A) < \beta$, $$ E \bigl ( \|| \xi |\| \cdot \chi_A \bigr ) ^k < \alpha \, . $$ \endproclaim \proclaim{Sublemma 3} (Essentially in Baxendale [1984].) Fix $k > 0$. Then $$ E \| | \xi_\epsilon - \text{Id} |\| ^k \to 0 \quad \text{as} \, \, \epsilon \to 0 \, . $$ \endproclaim \demo{Proof of Sublemma 3} Write $$ E \| | \xi_\epsilon - \text{Id} |\| ^k = \sum_{n=1}^\infty E \bigl ( \|| \xi_\epsilon - \text{Id} |\| \cdot \chi_{A_n} \bigr ) ^k \, . $$ First let $\epsilon \to 0$ for fixed $\delta$ to get $$ \lim_{\epsilon \to 0} E \|| \xi_\epsilon - \text{Id} |\| ^k \le \sup \{ \| | \xi - \text{Id} |\| \, : \, \xi \in \UU_\delta \} \, . $$ The quantity on the right clearly tends to zero as $\delta \to 0$. \qed \enddemo \subhead C. Statement of our results \endsubhead \smallskip \proclaim{Theorem 2} Let $f : M \to M$ be a $\CC^r$ expanding map as described in Section 4.A, with expanding constant $\lambda$, and let $\mu_0 = \rho_0\, dm$ be its unique absolutely continuous invariant probability measure. Let $\{ \XX^\epsilon, \epsilon > 0\}$ be a small random perturbation of $f$ of the type described in Section 4.B, with invariant probability measure $\mu_\epsilon = \rho_\epsilon \, dm$. Then: \roster \item The dynamical system $(f,\mu_0)$ is stochastically stable under $\XX^\epsilon$ in the space of $\CC^{r-1}$ functions, i.e., the $\CC^{r-1}$-norm of $\rho_\epsilon -\rho_0$ tends to zero as $\epsilon \to 0$. \endroster Let $\tau_0$ and $\tau_\epsilon$ be the rates of decay of correlations for $f$ and $\XX^\epsilon$ respectively, in the space of $\CC^{r-1}$ functions. If, in addition, $\tau_0 > \lambda^{-(r-1)}$, then: \roster \item[2] The rate of mixing for $f$ is robust, i.e., $\tau_\epsilon \to \tau_0$ as $\epsilon \to 0$. \endroster We show in fact that \roster \item[3] For each $\delta > 0$, outside of $\{ |z| \le \lambda^{-(r-1)} + \delta\}$, the spectrum of $\LL_\epsilon$ converges to that of $\LL$ as $\epsilon \to 0$. \endroster \endproclaim \remark{Remark} We conjecture that the correct scaling in $\epsilon$ for this kind of perturbation is $\| \rho_\epsilon -\rho_0\|_{r-2}= O(\sqrt \epsilon)$. \endremark \subhead D. Dynamical lemmas \endsubhead \smallskip The setting and all notations are as in Sections 4.A and B, and except for the scaling statement the two lemmas needed are identical to those in Section 3. Once again, they are: \proclaim{Lemma 6} For fixed $n \in \integer^+$ and $\varphi \in \CC^{r-1}$, $$ \| \LL^n_\epsilon \varphi - \LL^n\varphi \| \to 0 \quad \text{as} \, \, \epsilon \to 0 \, . $$ \endproclaim \smallskip \proclaim{Lemma 7} Let $\Lambda > \lambda^{-(r-1)}$ be given. Then there exists $N_0 \in \integer^+$ such that for all $n \ge N_0$ there exists $\epsilon(n) > 0$ such that for each $\epsilon < \epsilon(n)$, $$ \| \LL_\epsilon^n - \LL^n \| < \Lambda^n \, . $$ \endproclaim \smallskip We will use the proof of Lemma 7, with $r=2$, to illustrate how the analysis in Section 3.D can be adapted to the present setting. The other proofs are handled similarly. We use the random maps representation of $\XX^\epsilon$, i.e., we consider the probability space $(\Omega, \nu_\epsilon)$ where $\Omega$ can be identified with $\text{Diff }^{r}(M)$ and $\nu_\epsilon$ is the distribution of $\xi_\epsilon$. We let $\xi_\epsilon(\omega)$ denote the diffeomorphism corresponding to $\omega \in \Omega$, and write $f_\omega = \xi_\epsilon(\omega) \circ f$. Using the notation in Section 3.D, we have $f^n_{\vec \omega} = f_{\omega_n} \circ \cdots \circ f_{\omega_1}$ if $\vec \omega = (\omega_1, \ldots, \omega_n) \in \Omega^n$, and $$ (\LL^n_\epsilon \varphi) (x) = \int \cdots \int \nu_\epsilon(d \omega_1) \cdots \nu_\epsilon(d\omega_n) \, (\LL^n_{\vec \omega} \varphi) (x) \, , $$ where $$ (\LL^n_{\vec \omega} \varphi) (x) = \sum_{y : f^n_{\vec \omega} y = x} \varphi(y) \cdot {1 \over |\det D f^n_{\vec \omega} (y) |} \, . $$ Let $n$ be fixed for now. For local considerations we will assume that we are in Euclidean space. \smallskip \proclaim{Sublemma 4} $$ {d \over dx_i} (\LL^n_\epsilon \varphi) = \int \cdots \int \nu_\epsilon(d \omega_1) \cdots \nu_\epsilon(d\omega_n) \, {d \over dx_i} (\LL^n_{\vec \omega} \varphi) \, . $$ \endproclaim \demo{Proof of Sublemma 4} We fix $x \in M$, and write $$ {d \over dx_i} ( \LL^n_{\vec \omega} \varphi) (x) = \lim_{t \to 0} \Phi_t (\vec \omega) \, , $$ where $$ \Phi_t (\vec \omega) = {1\over t} \biggl \{ \bigl(\LL^n_{\vec \omega} \varphi \bigr ) (x+ t \, u_i) - \bigl ( \LL^n_{\vec \omega} \varphi \bigr ) (x) \biggr \} = {d \over dx_i} \bigl ( \LL^n_{\vec \omega} \varphi \bigr ) (x_t)\, , $$ for some $x_t$, where $u_i$ is the unit vector in the $i^{\text{th}}$ direction. Our assertion amounts to exchanging the order of the limit and integrals. To do that, we will produce $\Phi \in L^1(\Omega^n, \nu_\epsilon^n)$ with $|\Phi_t| \le |\Phi|$. Differentiating the expression for $\LL^n_{\vec \omega} \varphi$ above, we observe that $\frac {d} {dx_i} (\LL^n_{\vec \omega} \varphi) (x_t)$ is the sum of finitely many terms, each one of which is bounded in absolute value by a product of the form $$ C \cdot \| \varphi\|_1 \cdot \| | \xi_\epsilon (\omega_1) |\|^{k_1} \cdots \| | \xi_\epsilon (\omega_n) |\|^{k_n} \, , $$ where $C$ is a constant depending on $f$ and $n$, and $k_1, \ldots, k_n$ depend on $n$ and the dimension of $M$. We set $\Phi(\vec \omega)$ to be the corresponding sum. It follows from Sublemma 1 that $\Phi$ is integrable. Hence the Dominated Convergence Theorem applies. \qed \enddemo Consider first $\vec \omega =(\omega_1, \ldots, \omega_n)$ where $f^k_{\vec \omega}$ is $\CC^2$ very near $f^k$ for $1 \le k \le n$, say $\| f^k_{\vec \omega} -f^k \|_{2} < \delta$ for some $\delta > 0$. We assume $\delta$ is small enough so that the inverse branches of $f^n_{\vec \omega}$ are easily identified with those of $f^n$. Then the same argument as in Section 3.D, line by line, gives $$ \LL^n_{\vec \omega} \varphi = \LL^n \varphi \pm c_{n,\delta} \| \varphi\|_1 \, , $$ and $$ {d \over d x_i} (\LL^n_{\vec \omega} \varphi) = {d \over dx_i} (\LL^n \varphi) \pm c_{n,\delta} \| \varphi\|_1 \pm C \lambda^{-n} \| \varphi\|_1 \, . $$ The strategy of our proof is as follows: first we choose $n$ and then $\delta = \delta(n)$ so that for all $\vec \omega$ with the properties above, we have $$ \| \LL^n_{\vec \omega} \varphi - \LL^n \varphi \| \le {\Lambda'} ^{n} \| \varphi \| $$ for some $\lambda^{-(r-1)} < \Lambda' < \Lambda$. We then choose $\epsilon\ll \delta$ such that if $\Omega_0 := \{ \omega \, : \, \| f_\omega -f \|_{2} \ge \delta \}$, then $\nu_\epsilon \Omega_0$ is very small, small enough that these ``bad'' $\vec \omega$ do not contribute significantly to $\| \LL^n_\epsilon \varphi - \LL^n \varphi \|$. More precisely, let $$ \eqalign { \Omega^n_0 &:= \{ (\omega_1, \ldots, \omega_n) \, : \, \omega_i \notin \Omega_0 \, , \, \forall i \} \cr \text{and} \quad& \cr \Omega^n_j &:= \{ (\omega_1, \ldots, \omega_n) \, : \, \omega_j \in \Omega_0 \}\, . \cr } $$ First we consider the $\CC^0$-norm: $$ \eqalign { | \LL^n_\epsilon \varphi - \LL^n \varphi | &= |\int_{\Omega^n} d \nu_\epsilon^n (\vec \omega) \bigl (\LL^n_{\vec \omega} \varphi - \LL^n \varphi \bigr ) | \cr &\le \int_{\Omega_0^n} |\LL^n_{\vec \omega} \varphi - \LL^n \varphi | + \sum_{j=1}^n \int_{\Omega_j^n} \biggl ( |\LL^n_{\vec \omega} \varphi | + |\LL^n \varphi | \biggr ) \, . \cr } $$ The $\Omega_0^n$-term has been shown to be bounded above by $c_{n,\epsilon} \cdot \| \varphi \|_1$, and $$ \int_{\Omega_j^n} |\LL^n \varphi | \le \| \LL^n \| \cdot \| \varphi \|_1 \cdot \nu_\epsilon \Omega_0 \, , $$ the last factor of which can be made small as $\epsilon \to 0$. It remains to estimate $\int_{\Omega_j^n} |\LL^n_{\vec \omega} \varphi |$. Note that $\LL^n_{\vec \omega} \varphi$ is a sum of finitely many terms of the form $$ { \varphi(\cdot) \over |\det Df_{\omega_1} (\cdot) | \cdots | \det Df_{\omega_n} (\cdot) | } \, . $$ This expression is bounded above by $$ C \cdot |\varphi| \cdot \| | \xi_\epsilon(\omega_1) |\|^{k_1} \cdot \cdots \cdot \| | \xi_\epsilon(\omega_n) |\|^{k_n} \, . $$ Its integral over $\Omega_j^n$ is therefore bounded above by $$ C \cdot |\varphi| \cdot \biggl ( \prod_{i\ne j} E \| | \xi_\epsilon |\|^{k_i} \biggr ) \cdot E \biggl ( \| | \xi_\epsilon |\|^{k_j}\cdot \chi_{\Omega_0} \biggr )\, . $$ By Sublemma 2, the last factor can again be arranged to be arbitrarily small by choosing $\epsilon$ small. This proves $$ | \LL^n_\epsilon \varphi - \LL^n \varphi | \le c_{n,\epsilon} \cdot \| \varphi \|_1 \, . $$ A similar argument (see Sublemma 4) gives $$ | {d \over dx_i} \LL^n_\epsilon \varphi - {d \over dx_i} \LL^n \varphi | \le {\Lambda'}^{n} \| \varphi\|_1 + c_{n,\epsilon} \| \varphi\|_1 \le \Lambda^n \| \varphi\|_1 \, .\qed $$ \smallskip \subhead E. Proof of Theorem 2 \endsubhead \smallskip Use Section 2 and proceed as in Section 3.E. \smallskip \head 5. Piecewise expanding maps of the interval \endhead \subhead A. The unperturbed model \endsubhead \smallskip We consider here $f : I \to I$, where $I=[0,1]$ and $f$ is a continuous piecewise $\CC^2$, piecewise expanding map. More precisely, we assume that there exists a partition $0=a_0 < a_1 < \cdots < a_M =1$ of $I$ such that for each $i$, the restriction $f|_{[a_i, a_{i+1}]}$ can be extended to a $\CC^2$ map with $\min |f'| \ge \lambda > 1$. The $a_i$ are called the {\it turning points} of $f$. The continuity assumption on $f$ is imposed only for simplicity of exposition. One could replace it by piecewise continuity and consider left-hand and right-hand limits of the turning points. Recall that for $\varphi : I \to \real$, the total variation of $\varphi$ on an interval $[a,b]$ is defined to be $$ \varr_{[a,b]} \varphi = \sup \{ \sum_{i=0}^n |\varphi(x_{i+1}) -\varphi(x_i)| \, : \, n \ge 1 \, , a \le x_0 < x_1 < \ldots < x_n \le b \} \, . $$ We use $| \varphi|_1 := \int_I |\varphi|$ to denote the $L^1$-norm of $\varphi$ with respect to Lebesgue measure. Let $BV := \{ \varphi : I \to \complex \, : \, \varr_I \varphi < \infty \}$. One often considers the Banach space $(BV, \| \cdot \|)$ where $$ \| \varphi \| = \varr_I \varphi + | \varphi |_1 \, . $$ Let $\LL$ be the Perron-Frobenius operator associated with $f$ acting on $(BV, \| \cdot \|)$. The spectrum of $\LL$ in this setting has been studied by many people (Wong [1978], Hofbauer--Keller [1982], Rychlik [1983]). It has been shown that $\LL$ is quasi-compact, its spectral radius is equal to one, it has unity as an eigenvalue, and its essential spectral radius is equal to $$ \Theta = \lim_{n \to \infty} (\sup (1/|(f^n)'|)^{1/n} \le 1/\lambda\, . $$ (The derivative of $f$ is not well-defined at the turning points, but both limits $f'_+ (a_i) =\lim_{x \downarrow a_i} f'(x)$ and $f'_- (a_i) =\lim_{x \uparrow a_i} f'(x)$ exist; we replace implicitly each occurrence of $f'(a_i)$ by the maximum of these two limits.) Let $\rho_0$ be an eigenfunction for the eigenvalue $1$, with $|\rho_0|_1 =1$. Then $\rho_0$ is the density of an invariant probability measure $\mu_0$ for $f$. We {\it assume} that $f$ has no other absolutely continuous invariant probability measure, and that $f$ is weak mixing with respect to $\mu_0$. Under these assumptions, it has been shown that $1$ is the only point of $\sigma(\LL)$ on the unit circle, its generalized eigenspace is one-dimensional, and that $\tau_0 := \sup \{ |z| \, : \, z \in \sigma(\LL), z \ne 1 \}< 1$ measures the exponential rate of decay of correlations for functions in $BV$ (Hofbauer--Keller [1982], Keller [1984]). In our analysis to follow, it will be necessary for us to work with some other norms in $BV$. For $0 < \gamma \le 1$, we define $$ \| \varphi \|_\gamma = \gamma \cdot \varr_I \varphi + | \varphi|_1 \, . $$ Note that for any $ 0 < \gamma < \gamma'$ the norms $\|\cdot\|_\gamma$ and $\| \cdot \|_{\gamma'}$ are equivalent. \subhead B. Type of perturbation: convolutions \endsubhead \smallskip As in Section 3.B, we consider a small random perturbation $\XX^\epsilon$ of $f$ by convolution. Let us make the assumption that $f(I) \subset [\delta,1-\delta]$, for some $\delta>0$, so that we can avoid the problems at the boundary of $I$ when $f$ is perturbed. (There are other ways to deal with this.) We obtain as before a perturbed transfer operator $\LL_\epsilon$ acting on $(BV, \| \cdot \|)$. As in the first two models, $\LL_\epsilon$ has $1$ as an eigenvalue with eigenfunction $\rho_\epsilon$ which is the density of an invariant probability measure $\mu_\epsilon$ for $\XX^\epsilon$ (Lemma 19 in Keller [1982]). It is known that not all piecewise expanding maps are stochastically stable. A major difference between the situation here and that in Section 3 is that we do not have the kind of ``shadowing'' property used in the proof of Lemma 5. More precisely, let $\vec t = (t_1, \ldots, t_n)$ and $f^n_{\vec t}$ be as in Section 3.D. We count the smallest number of intervals on which $f^n$ is monotone, for that measures in some way the number of ``distinct orbits'' of $f$. In general $f^n_{\vec t}$ may have many more intervals of monotonicity than $f^n$. See Figure~1 for an example in which a turning point fixed by $f$ generates $2^n-2$ extra intervals of monotonicity for $f^n_{\vec t}$. This example is not stochastically stable, not even in the sense of weak convergence of $\mu_\epsilon$ (see Keller [1982, \S 6]). We remark that the ``shadowing'' property used in our proof of Lemma 5 is not the usual shadowing property: we deal only with orbits of finite length but require a complete matching of backwards branches of the map. For more information on the usual shadowing for interval maps see Coven--Kan--Yorke [1988]. \subhead C. Statement of our results \endsubhead \smallskip >From our discussion in the last subsection we see that our situation improves if the turning points do not get mapped near themselves. We say that $f$ has no {\it periodic turning point} if $f^{k} (a_i) \ne a_i$ for all $k \ge 1$. The kernel $\theta_\epsilon$ used in our convolutions is called symmetric if $\theta_\epsilon(x)=\theta_\epsilon(-x)$, $\forall x$. The definition of $\Theta$ is given in Section 5.A. We first state our result assuming that $f$ has no periodic turning points \figure 1 fig1.ps The fourth iterate of a map with a fixed turning point compared to the fourth iterate of a perturbation\cr \proclaim{Theorem 3} Let $f: I\to I$ be as described in Section 5.A, with a unique absolutely continuous invariant probability measure $\mu_0= \rho_0 \, dm$, and let $\XX^\epsilon$ be a small random perturbation of $f$ of the type described in Section 5.B with invariant probability measure $\rho_\epsilon \, dm$. We assume also that $f$ has no periodic turning points. Then \roster \item The dynamical system $(f,\mu_0)$ is stochastically stable under $\XX^\epsilon$ in $L^1(dm)$, i.e., $|\rho_\epsilon - \rho_0|_1$ tends to $0$ as $\epsilon \to 0$. \endroster Let $\tau_0$ and $\tau_\epsilon$ be the rates of decay of correlations for $f$ and $\XX^\epsilon$ respectively for test functions in $BV$. \roster \item[2] If $\tau_0^2 > \Theta$ then $\tau_\epsilon \to \tau_0$ as $\epsilon \to 0$. \endroster We show in fact that \roster \item[3] if we let $\tau = \min \{ |z| \, : \, z \in \sigma(\LL) \, , |z| > \sqrt\Theta \}$, then there exists $\delta > 0$ such that the spectrum of $\LL_\epsilon$ restricted to $\{ |z| \ge \tau - \delta\}$, converges to that of $\LL$ (restricted to the same domain) as $\epsilon \to 0$. \endroster \endproclaim \proclaim{Theorem 3'} Let $f$ and $\XX^\epsilon$ be as in Theorem 3, except that we do not require that $f$ has no periodic turning points. Then \roster \item is true if either $\Theta < 1/2$; or $\Theta < 2/3$ {\it and} $\theta_\epsilon$ is symmetric; \item and \therosteritem{3} are true if $\sqrt \Theta$ is replaced by $\sqrt{2 \Theta}$; or $\sqrt\Theta$ by $\sqrt{(3/2) \Theta}$ {\it if} $\theta_\epsilon$ is symmetric. \endroster \endproclaim The square roots arise from our use of balanced norms in the proofs of Lemmas 9 and 9'. We do not know to what extent they are needed. We do not know either if we can weaken the replacement of $\Theta$ by $2\Theta$ (or $(3/2) \Theta$) in Theorem 3'. However, it is clear that {\it some} hypothesis on $f$ or on the nature of our perturbations is necessary to give the type of results we want (see Section 5.B). We remark also that the hypothesis we use for proving stochastic stability is slightly weaker than that in Keller [1982,\S6] or Kifer [1988a, Chapter IV] (in the latter reference, only weak convergence is shown and the assumption that $\lambda > 2$ is implicitly used). \subhead D. Dynamical lemmas \endsubhead \smallskip The setting and notations are as in Sections 5.A and 5.B. We have the obvious lemma: \proclaim{Lemma 8} For fixed $n \ge 1$ and $\varphi \in L^1$ $$ | \LL^n_\epsilon \varphi - \LL^n \varphi|_1 \to 0 \; \; \text{as } \epsilon \to 0 \, . $$ \endproclaim \bigskip It is not true in general that $\varr( \LL_\epsilon \varphi - \LL \varphi)\to 0$ as $\epsilon \to 0$ for a fixed $\varphi \in BV$. We will use the notations $c_{n,\epsilon}$, $g = 1/|f'|$, and $f^n_{\vec t}$, $\LL^n_{\vec t}$ of Section 3.D. We also write $$ \eqalign { g^n(y)&= g(y) \cdot g(f y) \cdots g(f^{n-1} y)\cr g^n_{\vec t}(y_{\vec t})&= g(y_{\vec t}) \cdot g(f_{\vec t} y_{\vec t}) \cdots g( f^{n-1}_{\vec t} y_{\vec t}) \, .\cr } $$ We let $$ M_i := \# \{ k \, : \, k \ge 1, f^k (a_i) \in \{a_0, \ldots, a_M\} \} \, , \, $$ and $\MM = \max M_i\le M+1$. Note that $f$ is without periodic turning points if and only if $\MM < \infty$. Denote by $\ZZ_n$ the ``partition'' of $I$ into (closed) intervals of monotonicity of $f^n$, and by $\ZZ_{n,\vec t}$ the ``partition'' of $I$ into (closed) intervals of monotonicity for $f^n_{\vec t}$. Write $\ZZ_1 = \eta_1 \cup \ldots \cup \eta_M$. By definition an element $\eta(j_0,\ldots, j_{n-1})$ of $\ZZ_{n}$ is an interval of the form $$ \eta(j_0,\ldots, j_{n-1}) = \eta_{j_0} \cap f^{-1} (\eta_{j_1}) \cap \ldots \cap f^{-(n-1)} (\eta_{j_{n-1}}) \, , $$ with nonempty interior; and an element $\eta'(j_0,\ldots, j_{n-1})$ of $\ZZ_{n,\vec t}$ is an interval of the form $$ \eta'(j_0,\ldots, j_{n-1}) = \eta_{j_0} \cap f^{-1}_{(t_1)} (\eta_{j_1}) \cap \ldots \cap f^{-(n-1)}_{(t_1, \ldots t_{n-1})} (\eta_{j_{n-1}}) $$ with nonempty interior. \smallskip If $\MM=0$, it is not difficult to check that for fixed $n \ge 1$, there exists $\epsilon(n)$ such that, for all $\epsilon < \epsilon(n)$, the elements of $\ZZ_{n,\vec t}$ are in bijection with those of $\ZZ_n$. We say that two such intervals $\eta(j_0,\ldots, j_{n-1})\in \ZZ_n$ and $\eta'(j_0,\ldots, j_{n-1})\in \ZZ_{n,\vec t}$ are {\it associated} and that $\eta'$ is {\it admissible}. (Think of $\epsilon$ as being so small that two associated intervals are virtually the same.) Consider now the case $\MM \ge 1$. We fix $n$ and assume that $\epsilon$ is sufficiently small for this value of $n$. Consider $f^n_{\vec t}$ where each $|t_i| < \epsilon$. We associate elements of $\ZZ_n$ with those in $\ZZ_{n,\vec t}$ as before, but in general this will not account for all the elements of $\ZZ_{n,\vec t}$. An element of $\ZZ_{n,\vec t}$ without a counterpart in $\ZZ_n$ is called {\it nonadmissible}. Let us look at how nonadmissible elements are created. Let $a_i$ be a turning point, and let $q > 0$ be the first time $f^j a_i$ returns to the turning set. From the definition of $\ZZ_{q,\vec t}$, we see that the two intervals adjacent to $a_i$ in $\ZZ_{q,\vec t}$ are admissible, but that $\ZZ_{q+1,\vec t}$ may have two nonadmissible intervals adjacent to $a_i$. This is due to the fact that $f^q(a_i-\delta, a_i+\delta)$ lies on one side of some turning point $a_{i'}$, while $f^q_{\vec t}(a_i-\delta, a_i+\delta)$ may intersect both sides of $a_{i'}$. We think of these two newly created nonadmissible intervals as so short that their dynamics up to time $n$ is tied to that of $a_i$. If there is no $q'$, with $q < q'< n$, such that $f^{q'}a_i$ is in the turning set again, then in $\ZZ_{n,\vec t}$ these two nonadmissible intervals will be the only ones between the admissible intervals nearest to $a_i$. If, however, such a $q'$ exists, then the same mechanism as before may create two new nonadmissible intervals for $\ZZ_{q'+1, \vec t}$. In addition to that, each one of the already existent nonadmissible intervals near $a_i$ may get divided again, giving rise to a total of $2^2+2=6$ nonadmissible intervals near $a_i$ in $\ZZ_{q'+1,\vec t}$. Continuing this reasoning, if $a_i$ returns to the turning set $L$ times before time $n$, then the maximum number of nonadmissible intervals created near $a_i$ is $2^{(L+1)}-2$. Also, if $f^k x = a_i$ for $k < n$, then an imprint of the picture at $a_i$ is made at $x$, giving rise to other nonadmissible intervals between admissible ones in $\ZZ_{n,\vec t}$. These are the {\it only} ways in which nonadmissible intervals are created. To sum up, we have the following estimates. If $f$ has no periodic turning points, i.e. if $\MM < \infty$, then between any two admissible intervals in $\ZZ_{n,\vec t}$ there are at most $2^{\MM+1}$ nonadmissible ones. If $f$ has periodic turning points, then the maximum number of contiguous nonadmissible intervals is at most $2^n -2$. \smallskip We now ``trim'' the intervals of $\ZZ_n$ and the admissible intervals of $\ZZ_{n,t}$. Assume first that $\MM=0$ and $\epsilon$ is small enough. Let $\eta \in \ZZ_n$ and $\eta' \in \ZZ_{n,\vec t}$ be a pair of associated intervals of monotonicity. We decompose $\eta$ and $\eta'$ into two parts as follows: set $G(\eta,\eta') = f^n \eta \cap f^n_{\vec t } \eta'$ and $\eta_G = (f^n|_\eta)^{-1} (G)$, $\eta'_G = (f^n_{\vec t}|_\eta')^{-1} (G)$; and let $\eta_B = \eta \setminus \eta_G$ and $\eta'_B = \eta' \setminus \eta'_G$. We again say that the intervals $\eta_G$ and $\eta'_G$ are {\it associated} and that $\eta_B$ and $\eta'_B$ are their respective {\it co-respondents}. We denote by $B$ the union of all co-respondents $\eta_B$ and by $B'$ the union of all co-respondents $\eta'_B$. Then, for fixed $n$ the measures of $B$ and $B'$ both tend to zero as $\epsilon$ tends to zero. In the case where $\MM\ge 1$, we decompose associated intervals $\eta \in \ZZ_n$ and $\eta' \in \ZZ_{n,\vec t}$ into $\eta'=\eta'_{G} \cup \xi_{B}$ and $\eta=\eta_G\cup \eta_B$ as described in the case $\MM=0$. We again say that $\eta_G$ and $\eta'_G$ are {\it associated} and that $\eta_B$ is the {\it co-respondent} of $\eta_G$. We define the {\it co-respondents} of $\eta'_G$ to be $\xi_B$ together with half of the non-admissible intervals immediately to the left and half of those to the right of $\eta'$. Each non-admissible interval is hence the co-respondent of a unique $\eta'_G$. We denote by $B$ the union of all the ``bad'' intervals $\eta_B$, and by $B'$ the union of all co-respondents. \proclaim{Lemma 9} Assume that $f$ has no periodic turning points and let $\Theta < \Lambda^2 < 1$. Then, there exist $C > 0$ and $N_0 \in \integer^+$ such that for each $n \ge N_0$ there exists $\epsilon(n) > 0$ such that for each $\epsilon < \epsilon(n)$, $$ \| \LL_\epsilon ^n - \LL^n \|_{\Lambda^n} < C \cdot \Lambda^n \, . $$ \endproclaim Recall that $\| \cdot \|_{\Lambda^n}$ is the balanced norm with weight $\Lambda^n$. (See Section 5.A.) \demo {Proof of Lemma 9} In the proof, $\tilde \Theta$ denotes a generic constant slightly larger than $\Theta$. (We will have to increase $\tilde \Theta$ slightly a finite number of times in the argument.) There exists an $n_0$ such that $g^n(x) \le \tilde \Theta^n$ if $n \ge n_0$. \smallskip We have $$ \| \LL^n \varphi - \LL^n_{\vec t} \varphi \| \le \| \LL^n_{\vec t} (\varphi \chi_{B'})\| + \| \LL^n (\varphi \chi_{B})\| + \| \LL^n (\varphi \chi_{(I\setminus B)} - \LL^n_{\vec t} (\varphi \chi_{I\setminus B'})\| \, .\tag5.1 $$ We start with the details of the proof for the first ``bad'' term $\| \LL^n_{\vec t} (\varphi \chi_{B'})\|$; the second ``bad'' term is obtained by similar (more classical) bounds. The third term will be considered in Equations \thetag{5.10} to \thetag{5.14} below. For each $\eta'_B \in B'$ and for $x \in f^n_{\vec t} \eta'_B$, we have $$ \LL^n_{\vec t} (\varphi \chi_{\eta'_B}) (x)= \varphi(y_{\vec t}) \cdot g(y_{\vec t}) \cdots g(f^{n-1} y_{\vec t}) \, , $$ where $y_{\vec t}$ is the unique element of $\eta'_B$ such that $f^n_{\vec t} (y_{\vec t})=x$. It follows that $$ | \LL^n_{\vec t} (\varphi \chi_{\eta'_B})|_1 \le \int_{\eta'_B} |\varphi| \le \ell(\eta'_B) \cdot (\varr \varphi + | \varphi |_1 ) \, ,\tag5.2 $$ where $\ell(\eta'_B)$ denotes the length of the interval $\eta'_B$. Summing \thetag{5.2} over all intervals $\eta'_B$, we get $$ | \LL^n_{\vec t} (\varphi \chi_{B'})|_1 \le c_{n,\epsilon} \cdot (\varr \varphi + | \varphi|_1) \, . \tag5.3 $$ For the variation, we have for any interval $\eta' \in \ZZ_{n,\vec t}$ $$ \varr \LL^n_{\vec t} (\varphi \chi_{\eta'}) \le \varr_{\eta'} \varphi \cdot \sup_{\eta'} g^n_{\vec t} + \sup_{\eta'} |\varphi| \cdot \varr_{\eta'} g^n_{\vec t} + 2 \cdot \sup_{\eta'} |\varphi|\cdot \sup_{\eta'} g^n_{\vec t}\, . \tag5.4 $$ Were it not for the last term of \thetag{5.4}, everything would be much easier! We will use the following easily proved inequalities: if $n$ is large enough, say $n \ge n_1$, and $\epsilon$ is sufficiently small, then for $\eta'\in \ZZ_{n,\vec t}$ $$ \cases \sup_{\eta'} g^n_{\vec t} &\le \tilde \Theta^n \cr \varr_{\eta'} g^n_{\vec t} &\le \tilde \Theta^n \, . \cr \endcases\tag5.5 $$ (The first inequality is obvious, the second is proved by induction.) \document Set $n_2 = \max(n_0,n_1)$ and assume first that $n=n_2$. The interval $\eta'_B$ is a subset of some $\eta' \in \ZZ_{n,\vec t}$ and is a co-respondent of a unique good interval $\eta'_G$. From \thetag{5.4} and \thetag{5.5}, denoting by $\eta''$ the smallest interval containing $\eta'_G$ and $\eta'$, we obtain: $$ \eqalign { \varr \LL^{n}_{\vec t} (\varphi \chi_{\eta'_B}) &\le \varr \LL^{n}_{\vec t} (\varphi \chi_{\eta'}) \cr &\le \varr_{\eta'} \varphi \cdot \tilde \Theta^n + \biggl ( \varr_{\eta''} \varphi + \inf_{\eta''} | \varphi| \biggr ) \biggl ( \varr_{\eta'} g^{n}_{\vec t} + 2 \cdot \sup_{\eta'} g^{n}_{\vec t} \biggr ) \cr &\le \varr_{\eta''} \varphi \cdot 4 \tilde \Theta^n + \biggl ({ \varr_{\eta'} g^{n}_{\vec t} + 2 \cdot \sup_{\eta'} g^{n}_{\vec t} \over \ell(\eta'') } \biggr ) \cdot \ell(\eta'') \inf_{\eta''} | \varphi| \cr &\le \varr_{\eta''} \varphi \cdot 4 \tilde \Theta^{n} + D \cdot \ell(\eta'') \inf_{\eta''} | \varphi| \, , \cr } \tag5.6 $$ where $D =\sup_{\eta'\in \ZZ_{n,\vec t}} \bigl [ \varr_{\eta'} g^{n_2}_{\vec t} + 2 \cdot \sup_{\eta'} g^{n_2}_{\vec t} \bigr ]/ \ell_{n_2}$, with $\ell_{n_2}$ equal to the infimum of the lengths of admissible intervals in $\ZZ_{n_2,\vec t}$. Note that when $\epsilon$ tends to zero, $\ell_{n_2}$ tends to $\inf\ell(\eta)$, for $\eta$ in $\ZZ_{n_2}$, and observe that $\ell (\eta'') \inf_{\eta''} | \varphi|\le \int_{\eta''} |\varphi|$. Summing \thetag{5.6} over all intervals $\eta'_B$, and using the fact that the good intervals $\eta'_G$ are overcounted at most $2^\MM$ times, we get for $n=n_2$ $$ \varr(\LL^n_{\vec t} (\varphi \chi_{B'})) \le 4 \cdot 2^{\MM} \cdot \tilde \Theta^n \cdot \varr(\varphi) + 2 ^{\MM} D \cdot |\varphi|_1 \, , $$ and, by increasing $\tilde \Theta$ slightly and assuming $n_2$ is large enough, $$ \eqalign { \varr(\LL^n_{\vec t} (\varphi \chi_{B'})) &\le \sum_{\eta'\in \ZZ_{n,\vec t}} \varr(\LL^n_{\vec t} (\varphi \chi_{\eta'})) \cr &\le \tilde \Theta^n \cdot \varr(\varphi) + 2 ^{\MM} D \cdot |\varphi|_1 \, . } \tag 5.7 $$ If $n > n_2$, write $n = q \cdot n_2 + r$ with $r < n_2$. If a vector $\vec t$ of length $2 n_2$ is the concatenation of two vectors $\vec u$ and $\vec v$ of length $n_2$, and $\xi, \zeta$ are the unique intervals in $\ZZ_{n_2, \vec u}$, respectively $\ZZ_{n_2,\vec v}$ such that a given $\eta' \in \ZZ_{n,\vec t}$ is equal to $(f^{n_2}_{\vec v}|_{\xi})^{-1} (\xi) \cap \zeta$ then $$ \LL^{2 n_2}_{\vec t} (\varphi \chi_\eta') = \LL^{n_2}_ {\vec u} (\chi_\xi \cdot \LL^{n_2}_{\vec v} (\chi_\zeta \cdot \varphi)) \, . $$ In particular $$ \eqalign { \varr \sum_{\xi \in \ZZ_{n_2, \vec u}} \LL^{n_2}_{\vec u} (\chi_\xi \cdot \LL^{n_2}_{\vec v} (\chi_\zeta \cdot \varphi) ) &\le \tilde \Theta^{n_2} \cdot \varr(\LL^{n_2}_{\vec v} (\chi_\zeta \cdot \varphi) ) + 2 ^{\MM} D \cdot |\LL^{n_2}_{\vec v} (\chi_\zeta \cdot \varphi)|_1 \cr &\le \tilde \Theta^{n_2} \cdot \varr(\LL^{n_2}_{\vec v} (\chi_\zeta \cdot \varphi) ) + 2 ^{\MM} D \cdot \int_\zeta |\varphi| \, . \cr } $$ A standard induction argument yields $$ \varr \LL^{n}_{\vec t} (\varphi \chi_{B'}) \le \tilde \Theta^n \cdot \varr \varphi + D' \cdot | \varphi|_1 \, , \tag5.8 $$ where $D'$ is essentially $2^\MM D/(1-\tilde \Theta)$ (see e.g. Rychlik [1983, Lemma 7, and Proposition 1]). The problem we have to deal with now is that the term $D' \cdot |\varphi|_1 $ in \thetag{5.8} is not small. To do this, we follow the ``balancing'' idea suggested to us by Collet [1991]. Not knowing which $\gamma$ to choose for now, we rewrite \thetag{5.3} and \thetag{5.8} using our new norm $\| \cdot \|_\gamma$: $$ \eqalign { | \LL^n_{\vec t} (\varphi \chi_{B'})|_1 &\le c_{n,\epsilon}\cdot (\gamma \cdot \varr \varphi + | \varphi|_1) \, , \cr \gamma \cdot \varr(\LL^n_{\vec t} (\varphi \chi_{B'})) &\le \gamma \cdot \tilde \Theta^n \cdot \varr \varphi + \gamma\cdot D' \cdot |\varphi|_1 \, . } $$ Together, they give $$ \| \LL^n_{\vec t} (\varphi \chi_{B'})\|_\gamma \le ( c_{n,\epsilon} + \tilde \Theta^n + \gamma \cdot D' ) \cdot \|\varphi\|_\gamma \le (\tilde \Theta^n + D' \cdot \gamma)\cdot \| \varphi\|_\gamma \, .\tag5.9 $$ \medskip \bigskip We now bound the difference $\| \LL^n \varphi \chi_{(I\setminus B)}- \LL^n_{\vec t} (\varphi \chi_{(I\setminus B')})\|$. We first consider the supremum norm to control the $L^1$ part. Let us fix some point $x$ in $f^n_{\vec t} (I\setminus B')$. By assumption, there exist two nonempty sets of intervals $\eta'_{G,j} \subset I \setminus B'$, and $\eta_{G,j} \subset I \setminus B$ ($j=1, \ldots k(x)$) such that $x \in f^n(\eta_{G,j})=f^n_{\vec t} (\eta'_{G,k})$ for $j=1, \ldots k(x)$. Fixing $j$ and denoting by $y$, respectively $y_{\vec t}$, the unique $n$-preimage of $x$ in $\eta=\eta_{G,j}$, respectively $\eta'=\eta'_{G,j}$, we have $d(y,y_{\vec t}) = c_{n,\epsilon}$ and hence $$ \eqalign{ \LL^n_{\vec t}(\varphi \chi_{\eta'}) (x) &= \varphi(y_{\vec t}) g(y_{\vec t}) \ldots g (f^{n-1} y_{\vec t}) \cr &\le (\varphi(y) + \varr_{\eta \cup \eta'} \varphi) \cdot (g^n(y) + c_{n,\epsilon}) \cr &\le \LL^n(\varphi \chi_{\eta}) (x) + \tilde \Theta^n \cdot \varr_{\eta \cup \eta'} \varphi + c_{n,\epsilon} \cdot (\varr_{\eta \cup \eta'} \varphi + \sup | \varphi|) \, . }\tag 5.10 $$ We have an analogous lower bound. Summing over $j$, we get: $$ \eqalign { |\LL^n_{\vec t} (\varphi \chi_{(I\setminus B')}) - \LL^n (\varphi\chi_{(I\setminus B)} ) |_1 &\le \sup |\LL^n_{\vec t} (\varphi \chi_{(I\setminus B')}) (x) - \LL^n (\varphi\chi_{(I\setminus B)} ) (x)|\cr &\le \tilde \Theta^n \varr \varphi + c_{n, \epsilon} | \varphi |_1 \, . } \tag5.11 $$ The ``trimming'' was not really needed for the bound \thetag{5.11} on the $L^1$-norm since $f^n(B) \cup f^n_{\vec t} (B')$ has a measure tending to zero as $\epsilon$ tends to zero, but it will be crucial for the next bound. Consider an associated pair $(\eta_G,\eta'_G)$ which for simplicity of notation we write as $(\eta,\eta')$. Defining the bijection $\Psi:\eta' \to \eta$ by $\Psi(y_{\vec t}) = y$, we obtain $$ \eqalign { \varr (\LL^n_{\vec t} (\varphi \chi_{\eta'})&- \LL^n (\varphi \chi_{\eta}) ) = \varr (g^n_{\vec t} \varphi \chi_{\eta'} - (g^n \varphi ) \circ \Psi \chi_{\eta'} ) \cr &\le \varr ((g^n _{\vec t} \varphi \chi_{\eta'}- g^n_{\vec t} (\varphi \circ \Psi) \chi_{\eta'}) +\varr(g^n_{\vec t} (\varphi \circ \Psi) \chi_{\eta'}- (g^n \varphi ) \circ \Psi \chi_{\eta'}) \cr &\le \varr_{\eta'} (g^n_{\vec t} (\varphi -\varphi \circ \Psi)) + \varr_{\eta=\Psi {\eta'}} (\varphi (g^n_{\vec t}\circ \Psi^{-1} - g^n ))\cr &\quad+ 2 \sup_{\eta'} (g^n_{\vec t} (\varphi -\varphi \circ \Psi)) +2 \sup_{\eta} (\varphi (g^n_{\vec t}\circ \Psi^{-1} - g^n ))\cr &\le \sup_{\eta'} g^n_{\vec t} \cdot \varr_{\eta'} (\varphi - \varphi \circ \Psi) + \varr_{\eta'} g^n_{\vec t} \cdot \sup_{\eta'} |\varphi - \varphi \circ \Psi|\cr &\quad+ \sup_{\eta} |\varphi| \cdot \varr_{\eta'} (g^n_{\vec t} -g^n \circ \Psi) + \varr_{\eta} \varphi \cdot \sup_{\eta'} |g^n_{\vec t} -g^n \circ \Psi| \cr &\quad+ 2 \sup_{\eta'} g^n_{\vec t} \cdot \sup_{\eta'} |\varphi - \varphi \circ \Psi| + 2 \sup_{\eta} |\varphi| \cdot \sup_{\eta'} |g^n_{\vec t} -g^n \circ \Psi| \cr &\le 2 \tilde \Theta^n \cdot \varr_{\eta'\cup \eta} (\varphi) + \tilde \Theta^n \cdot \varr_{\eta\cup\eta'} \varphi + \sup_{\eta} |\varphi| \cdot c_{n,\epsilon} + \varr_{\eta} \varphi \cdot 2 \tilde \Theta^n \cr &\quad+ 2 \tilde \Theta^n \cdot \varr_{\eta\cup\eta'} \varphi + 2 \sup_\eta {|\varphi|}\cdot c_{n,\epsilon} \, , \cr }\tag5.12 $$ where we have used that $f$ is $\CC^2$ in the last inequality to get $\varr_{\eta'} (g^n_{\vec t} -g^n \circ \Psi) \le c_{n,\epsilon}$. We have also used the fact that $\eta' \to \eta$ as $\epsilon \to 0$, so that $\eta \cup \eta'$ is a connected interval. Summing the above inequalities over all elements of $\ZZ_n$, and noting that intervals of the form $\eta\cup \eta'$ intersect at most two of their neighbors, we get $$ \eqalign { \varr (\LL^n (\varphi \chi_{I \setminus B}) - \LL^n_{\vec t} (\varphi \chi_{I \setminus B'})) &\le \tilde \Theta^n \cdot (\varr \varphi + |\varphi|_1) \, .\cr }\tag5.13 $$ >From \thetag{5.11} and \thetag{5.13} we find: $$ \eqalign { \| \LL^n (\varphi \chi_{I\setminus B}) - \LL^n_{\vec t} (\varphi \chi_{I\setminus B'})\|_\gamma &\le \gamma^{-1} \cdot \tilde \Theta^n \cdot (\varr \varphi + | \varphi|_1 )\cr &\le \gamma^{-1} \cdot \tilde \Theta^n \cdot \|\varphi\|_\gamma \, . } \tag5.14 $$ Adding \thetag{5.9}, the analogue of \thetag{5.9} for $\LL^n$ and \thetag{5.14}, and integrating over $\vec t$, we obtain $$ \| \LL_\epsilon^n - \LL^n \|_\gamma \le 2 (\tilde \Theta^n + D' \gamma) + \gamma^{-1} \tilde \Theta^n \, . $$ Remembering that $\Lambda^2 > \tilde \Theta$, we see that if we let $\gamma = \Lambda^n$, then the right side of the above inequality is bounded above by $C\cdot \Lambda^n$. This completes the proof of Lemma 9.\qed \enddemo \smallskip \proclaim {Lemma 9'} Let $\Lambda$ be such that $\Theta < \min (\Lambda,2 \Lambda^2)$. Then there exist $C > 0$ and $N_0 \in \integer^+$ such that for each $n \ge N_0$ there exists $\epsilon(n) > 0$ such that for each $\epsilon < \epsilon(n)$, $$ \| \LL_\epsilon ^n - \LL^n \|_{\Lambda^n} < C \cdot(2 \Lambda)^n \, . $$ If each $\theta_\epsilon$ is symmetric, then for $\Lambda$ such that $\Theta < \min (\Lambda, (3/2) \Lambda ^2)$ we have the better inequality $$ \| \LL_\epsilon ^n - \LL^n \|_{\Lambda^n} < C \cdot({3\over2} \Lambda)^n \, . $$ \endproclaim \demo{Proof of Lemma 9'} We shall follow the proof of Lemma 9, noting only the modifications which are necessary when $\MM=\infty$. We see that the only important change occurs when we sum \thetag{5.6} over the intervals $\eta'_B$. Since each good interval $\eta'_G$ has at most $2^{n-1}$ co-respondents, the sum yields for $n=n_2$: $$ \varr(\LL^n_{\vec t} (\varphi \chi_{B'})) \le 2 \cdot (2 \tilde \Theta)^n \cdot \varr(\varphi) + 2 ^{n-1} D \cdot |\varphi|_1 \, . $$ For general $n=q\cdot n_2 + r$, the same induction argument as in the proof of Lemma 9 allows us to replace Inequality \thetag{5.8} by $$ \varr \LL^{n}_{\vec t} (\varphi \chi_{B'}) \le(2 \tilde \Theta)^n\cdot \varr \varphi + 2^n \cdot D' \cdot | \varphi|_1 \, . $$ Inequality \thetag{5.9} hence becomes $$ \| \LL^n_{\vec t} (\varphi \chi_{B'})\|_\gamma \le ( c_{n,\epsilon} + (2 \tilde \Theta)^n + \gamma \cdot 2^n \cdot D') \cdot \|\varphi\|_\gamma \, . $$ Inequality \thetag{5.14} does not have to be changed. Summing up, we have $$ \| \LL^n _\epsilon \varphi - \LL^n \varphi\|_\gamma \le \biggl ( (2 \tilde \Theta)^n + \gamma^{-1} \tilde \Theta^n + \gamma 2^n D' \biggr ) \| \varphi \|_\gamma \, , $$ and hence the inequality as claimed. \smallskip Assume now that each $\theta_\epsilon$ is symmetric. Again inequality \thetag{5.14} does not have to be changed, and it suffices to get a bound replacing \thetag{5.9}. Let $\eta'_G$ be a trimmed admissible interval for $f^n_{\vec t}$ which is associated with $\eta_G \subset \eta \in \ZZ_n$, where a boundary point $b$ of $\eta$ is periodic. We claim that there exists a sequence $S=\{ s_j\}_{j=1, \ldots n}$ of signs $s_j = \in \{+, -\}$ such that $\eta'_G$ has at most $2^{k(S)}$ nonadmissible co-respondents $\eta'_B$, where $0 \le k(S) \le n$ is the numbers of coordinates $t_i$ of $\vec t$ such that the sign of $t_j = s_j$. Indeed, take $s_j$ to be $+$ or $-$, depending on whether the $j^{\text{th}}$ iterate of $b$ is a local maximum or a local minimum respectively for $f^n$. (For example, in the map of Figure~1, the sequence of signs is $s_j = +$ for all $j$.) We first sum \thetag{5.6} over the bad intervals $\eta'_B$ for which $k(\eta'_B)$ is equal to some fixed $k$ and call this partial sum $A_k$. Since $\theta_\epsilon$ is symmetric, we have $$ \int \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n) A_k \le \binom { n} {k} {2^k \over 2^n} \cdot \bigl ( \tilde \Theta^n \cdot \varr \varphi + D \cdot | \varphi|_1 \bigr )\, , $$ hence, using $\sum_{k=1}^n \binom {n} {k} 2^k = 3^n -1$, $$ \varr \LL^{n}_\epsilon (\varphi \chi_{B'}) \le \sum_k \int \theta_\epsilon(t_1) \ldots \theta_\epsilon(t_n) A_k \le ((3/2) \cdot \tilde\Theta)^n \cdot \varr \varphi + (3/2)^n D \cdot | \varphi|_1 \, . $$ We thus obtain $$ \eqalign { \| \LL^n_\epsilon (\varphi \chi_{B'})\|_\gamma &\le ( c_{n,\epsilon} + ((3/2) \cdot \tilde \Theta)^n + \gamma \cdot (3/2)^n \cdot D' ) \cdot \|\varphi\|_\gamma\cr &\le \biggl [((3/2) \tilde \Theta)^n + \gamma \cdot (3/2)^n D' \biggr ] \cdot \| \varphi\|_\gamma \, , \cr } $$ which yields the claim. \qed \enddemo \smallskip We have implicitly used the following inequality in the proofs of Lemma 9 and Lemma 9': assume that $\psi(x, t)$ is a function of two variables such that the function $t \mapsto \theta_\epsilon(t) \psi(x, t)$ is in $L^1(dm)$ for each fixed $x$, then $$ \eqalign { | \int dt \, \theta_\epsilon(t) \psi (\cdot,t) |_1 &\le \int dt \, \theta_\epsilon(t) | \psi(\cdot,t)|_1 \cr \varr_x \bigl ( \int dt \, \theta_\epsilon(t) \psi (x,t) \bigr ) &\le \int dt \, \theta_\epsilon(t) \varr_x \psi(x,t) \, . \cr } $$ \smallskip As in the first two models, we have not used in the proofs the expanding condition as stated, but only the slightly weaker assumption $\Theta < 1$. \subhead E. Perturbation lemmas for abstract operators: a modified version of \S 2 \endsubhead \smallskip Because of the need to introduce the norms $\| \cdot \|_\gamma$, we need a slightly refined version of Section 2. Again, $(X , \| \cdot \|)$ is a complex Banach space, and $\{ T_\epsilon, \epsilon \ge 0 \}$ is a family of bounded linear operators on $X$. We assume that $T_0$ satisfies conditions (A.1) and (A.3) in Section 2, i.e., $\sigma(T_0) = \Sigma_0 \cup \Sigma_1$ with $$ \kappa_1 := \sup \{ |z| \, :\, z \in \Sigma_1 \} < \inf \{ |z| \, : \, z \in \Sigma_0 \} =: \kappa_0 \, , $$ and $\dim X_0 < \infty$. We further assume that $\Sigma_1$ can be written as the union of isolated sets $$ \Sigma_1 = \Sigma_{1,0} \cup \Sigma_{1,1} \, , \tag{A'.1} $$ where $\Sigma_{1,0}$ could be empty and $\dim X_{1,0}$ is at most finite. (The notations $\pi_{1,0}$, $\pi_{1,1}$, $X_{1,0}$ and $X_{1,1}$ have the obvious meanings.) Let $$ \kappa_{11} := \sup \{ |z| \, : \, z \in \Sigma_{1,1} \} \, . $$ We assume that there is another norm $| \cdot |$ on $X$ such that $|x | \le \| x \|$ for all $x$, and a family of norms $\| \cdot \|_\gamma$, with $0 < \gamma \le 1$ with $$ \| \cdot \|_\gamma = \gamma \| \cdot \| + (1-\gamma) | \cdot | \, . $$ (In particular $\gamma \| \cdot \| \le \| \cdot \|_\gamma \le \| \cdot \|$ and $|\cdot | \le \| \cdot \|_\gamma$.) Condition \thetag{A.2} is replaced by the assumption that there exists $\kappa$ with $(\kappa_{11}/\kappa_0 ) < \kappa < \kappa_0$ such that for each large enough $N \in \integer^+$ there exists $\epsilon(N)$ such that for all $0 < \epsilon < \epsilon(N)$ $$ \| T_\epsilon^N -T_0^N \|_{\kappa^N} \le \kappa^N \, . \tag{A'.2} $$ We shall need two sublemmas: \proclaim{Sublemma 5} Assume \thetag{A.1}, \thetag{A'.1}, and \thetag{A.3}. Then for any $\kappa_0'< \kappa_0$, $\kappa'_1 > \kappa_1$, there exists $N_0$ such that for all $n \ge N_0$, any $0 < \gamma\le 1$, and any $x \in X_0$, $y \in X_{1,0}$ \roster \item $\| T_0^n x \|_\gamma \ge (\kappa'_0)^n \| x\|_\gamma$, \item $\| T_0^n y \|_\gamma \le (\kappa'_1)^n \| y\|_\gamma$. \endroster \endproclaim \demo{Proof of Sublemma 5} We prove \therosteritem{1}. Since $X_0$ is finite dimensional, all norms are equivalent. We choose $N_0$ such that for all $n \ge N_0$ and $x \in X_0$, $$ |T_0^n x| \ge (\kappa'_0)^n |x| \quad \text{ and } \quad \|T_0^n x\| \ge (\kappa'_0)^n \|x\| \, . $$ The same inequality then holds for $\| \cdot \|_\gamma$ which is a weighted average of $|\cdot |$ and $\| \cdot \|$. \qed \enddemo \proclaim{Sublemma 6} If \thetag{A.1}, \thetag{A'.1}, and \thetag{A.3} hold, then there exists a constant $C$ such that for any $0 < \gamma \le 1$, we have $\|\pi_0\|_\gamma \le C$, $\|\pi_{1,0}\|_\gamma \le C$, and $\|\pi_1\|_\gamma \le 2 C +1$. \endproclaim \demo{Proof of Sublemma 6} For $x \in X$, we have $$ \| \pi_0 x \|_\gamma \le \| \pi_0 x \| \le \text{const} | \pi_0 | \cdot | x | \le \text{const} |\pi_0 |\cdot \|x \|_\gamma \, , $$ where we have used again the fact that the norms $| \cdot |$ and $\| \cdot \|$ are equivalent on the finite-dimensional space $X_0$. We proceed in the same way for $\|\pi_{1,0}\|_\gamma$. To finish, observe that $\pi_0 + \pi_{1,0} +\pi_1= I$ so that $\|\pi_1\|_\gamma \le \| \pi_0\|_\gamma + \|\pi_{1,0}\|_\gamma+ 1$. \qed \enddemo We can now prove: \proclaim{Lemma 1'} Assume \thetag{A.1},\thetag{A.3},\thetag{A'.1}, and \thetag{A'.2}, then the conclusion of Lemma 1 from Section 2 is true. \endproclaim \demo{Proof of Lemma 1'} Let $$ \eqalign { \kappa_1 < \kappa'_1 &< \kappa' < \kappa'_0 < \kappa''_0 < \kappa_0\, , \cr \kappa_{11} & < \kappa_{11}' <\kappa < \kappa' \, , \cr {\kappa_{11}' \over \kappa} & < \kappa' \, . \cr } $$ Let $N$ be large enough for various purposes. In particular, we require (see Sublemma 5) that $$ \eqalign { x \in X_0 &\Rightarrow \| T^N_0 x \|_{\kappa^N} \ge (\kappa''_0)^N \| x \|_{\kappa^N} \, , \cr x \in X_{1,0} &\Rightarrow \| T^N_0 x \|_{\kappa^N} \le (\kappa'_1)^N \| x \|_{\kappa^N} \, , \cr x \in X_{1,1} &\Rightarrow \| T^N_0 x \| \le (\kappa'_{11})^N \| x \| \, . \cr } $$ We let $\epsilon < \epsilon (N)$ and will show that $\lambda \notin \sigma(T_\epsilon)$ for $\lambda$ with $\kappa' < |\lambda| < \kappa'_0$ (if $\kappa'$ is close enough to $\kappa_0$). We proceed as in Lemma 1, using $\| \cdot \|_{\kappa^N}$ in the place of $\| \cdot \|$ and estimating $\| R(T_0^N, \lambda^N)\|_{\kappa^N}$ by projecting onto $X_0$, $X_{1,0}$ and $X_{1,1}$. It follows from our choice of constants that for $x \in X_0$, we have $$ \| T_0^N x -\lambda x \|_{\kappa^N} \ge \text{const} \cdot (\kappa''_0)^N \| x \|_{\kappa^N} \, , $$ and for $x \in X_{1,0}$, we have $$ \| T_0^N x -\lambda x \|_{\kappa^N} \ge \text{const} \cdot (\kappa')^N \| x \|_{\kappa^N} \, . $$ As for $x \in X_{1,1}$, we have $$ \| T^N_0 x \|_{\kappa^N} \le \|T_0^N x\| \le (\kappa'_{11})^N \| x\| \le ({\kappa'_{11}\over \kappa})^{N} \| x \|_{\kappa^N} \, , $$ from which it follows that $$ \| T_0^N x -\lambda x \|_{\kappa^N} \ge \text{const} \cdot (\kappa')^N \| x \|_{\kappa^N} \, . $$ These estimates together with Sublemma 6 give $$ \| R(T_0^N, \lambda^N) \|_{\kappa^N} \le {1\over \kappa^N} \, .\qed \tag5.15 $$ \enddemo Note that, unlike the situation in Section 2, $\kappa'$ cannot be taken arbitrarily near $\kappa$. Lemma 2 from Section 2 holds in the present setting, with convergence in the sense of the $\| \cdot \|_{\kappa^N}$-norm (i.e., for any $\delta > 0$ there are $N \in \integer^+$ and $\epsilon(N)$ such that, for each $\epsilon < \epsilon(N)$, $\|\pi_0 -\pi_0^\epsilon\|_{\kappa^N} < \delta$), and the same proof. Define $$ C_1^*(\epsilon) := \sup \Sb x \in X_0 \\ x \ne 0 \endSb { |T_\epsilon x - T_0 x | \over | x| } $$ and assume that $$ C_1^*(\epsilon) \to 0 \text { as } \epsilon \to 0 \, . \tag{A'.4} $$ \proclaim{Lemma 3'} Assume \thetag{A.1},\thetag{A.3},\thetag{A'.1}, \thetag{A'.2}, \thetag{A'.4}, and that $| T_\epsilon|$ is uniformly boun\-ded. Then $$ \sigma(T_\epsilon|_{X_0^\epsilon}) \to \sigma(T_0|_{X_0}) $$ as $\epsilon \to 0$. \endproclaim \smallskip \demo{Proof of Lemma 3'} As in Lemma 3, we show that $X_0^\epsilon=\text{graph} (S_\epsilon)$ for some linear $S_\epsilon : X_0 \to X_1$ with $\| S_\epsilon\|_{\kappa^N} \to 0$ as $N \to \infty$ and $\epsilon \to 0$, $\epsilon < \epsilon(N)$. Define $\hat T_\epsilon : X_0 \to X_0$ as before. To prove our claim, it suffices to show that $|\hat T_\epsilon - T_0| \to 0$ as $\epsilon \to 0$. Now for $x \in X_0$, with $|x|=1$, $$ \eqalign { | \hat T_\epsilon x - T_0 x | &\le |\pi_0| \cdot (|T_\epsilon x - T_0 x| + | T_\epsilon S_\epsilon x | ) \cr &\le | \pi_0| \cdot ( C_1^*(\epsilon)+ | T_\epsilon| \cdot |S_\epsilon x|)\, , } $$ and it only remains to show that $|S_\epsilon x| \to 0$. This is true because $$ \eqalign { |S_\epsilon x| &\le \| S_\epsilon \|_{\kappa^N} \| x \|_{\kappa^N}\cr &\le \| S_\epsilon \|_{\kappa^N} (\kappa^N \cdot \text{const} \cdot |x| + (1-\kappa^N)\cdot |x| )\, .\qed \cr } $$ \enddemo \demo{Proof of Theorem 3} Obviously we wish to apply the results above to $T_0=\LL$, $T_\epsilon=\LL_\epsilon$, $X=BV$ etc. We will indicate how to prove assertion \therosteritem{3}. Let $\Theta < \Theta' < \Theta''$ be such that $\Theta''$ is arbitrarily near $\Theta$. We let $$ \eqalign { \Sigma_{1,1} &= \{ z \in \sigma(\LL) \, : \, |z| \le \Theta' \} \, ,\cr \Sigma_{1,0} &= \{ z \in \sigma(\LL) \, : \, \Theta' < |z| < \sqrt{ \Theta''} \} \, ,\cr \Sigma_0 &= \{ z \in \sigma(\LL) \, : \, |z| \ge \sqrt {\Theta''} \} \, ,\cr } $$ and choose $\kappa=\Lambda$ near $\sqrt {\Theta''}$ such that $\Theta' < \kappa^2 < \kappa \sqrt {\Theta''}$. The norm of $\LL_\epsilon :L^1 \to L^1$ is equal to $1$, and it follows from Lemma 9 that $\LL_\epsilon$ is quasi-compact so that $\rho_\epsilon \in BV$ (see e.g. Keller [1982, p. 315]). Theorem 3 hence follows from Lemma 3' and the results stated in Sections 5.A and 5.B. (The fact that the $L^1$-norm is strictly speaking only a norm when one quotients out functions of bounded variation $\varphi$ for which $|\varphi|_1 =0$ is not a problem, see Proposition 1 in Baladi--Keller [1990].) \qed \enddemo \smallskip \demo{Proof of Theorem 3'} Again we prove \therosteritem{3}. We let $\kappa_{11} \le \Theta'$ be as above. Here, however, we consider only $\kappa_0> \sqrt{2 \Theta''}$ and let $\kappa=\Lambda$ be very slightly smaller than $\kappa_0/2$. Then $\Theta < 2 \kappa^2< \kappa$ which is the hypothesis of Lemma 9'. Lemma 9' does not yield \thetag{A'.2} but only the weaker bound $$ \| T_\epsilon^N -T_0^N \|_{\kappa^N} \le (2\kappa)^N \, . $$ However, since we can assume that the constant $\kappa'$ in the proof of Lemma 1' satisfies $\kappa_{11} < \kappa < 2 \kappa < \kappa' < \kappa_0$, we obtain an improved version of \thetag{5.15}: $$ \| R(T_0^N, \lambda^N) \|_{\kappa^N} \le { \text{const} \bigl ( \|\pi_0\|_{\kappa^N} + \|\pi_{1,0}\|_{\kappa^N} + \|\pi_1\|_{\kappa^N}\bigr ) \over (\kappa')^N} \le {1\over (2 \kappa)^N} \, . $$ The other requirement on $\kappa$ in \thetag{A'.2}, namely that $\kappa_{11} < \kappa \kappa_0$, is also satisfied. The conclusion of Lemma 1' is thus still valid. (The proof of Lemma 2 can be modified in a similar fashion.) We finish as in Theorem 3. If the functions $\theta_\epsilon$ are symmetric, we can replace each factor $2$ by $3/2$ in the above choices. \qed \enddemo \bigskip \Refs \ref \no 1 \by V. Baladi and G. Keller \paper Zeta functions and transfer operators for piecewise monotone transformations \jour Comm. Math. Phys. \yr 1990 \vol 127 \pages 459--477 \endref \ref \no 2 \by P. Baxendale \paper Brownian motions in the diffeomorphism group \jour Compositio Math. \vol 53 \yr 1984 \pages 19--50 \endref \ref \no 3 \by M. Benedicks and L.-S. Young \paper Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps \jour Ergodic Theory Dynamical Systems \yr 1992 \vol 12 \pages 13--37 \endref \ref \no 4 \by P. Collet \paper Ergodic properties of some unimodal mappings of the interval \paperinfo Preprint Mittag-Leffler (1984) \endref \ref \no 5 \bysame \paper Some ergodic properties of maps of the interval \inbook Dynamical Systems and Frustrated Systems \toappear \eds R. Bamon, J.-M. Gambaudo and S. Martinez \yr 1991 \endref \ref \no 6 \by P. Collet and S. Isola \paper On the essential spectrum of the transfer operator for expanding Markov maps \jour Comm. Math. Phys. \vol 139 \yr 1991 \pages 551--557 \endref \ref \no 7 \by E.M. Coven, I. Kan and J.A. Yorke \paper Pseudo-orbit shadowing in the family of tent maps \jour Trans. Amer. Math. Soc. \yr 1988 \vol 308 \pages 227--241 \endref \ref \no 8 \by J. Franks \paper Manifolds of $\CC^r$ mappings and applications to differentiable dynamical systems \jour Studies in Analysis, Adv. Math. Suppl. Stud. \yr 1979 \vol 4 \pages 271--291 \endref \ref \no 9 \by F. Hofbauer and G. Keller \paper Ergodic properties of invariant measures for piecewise monotonic transformations \jour Math. Z. \vol 180 \pages 119--140 \yr 1982 \endref \ref \no 10 \by G. Keller \paper Stochastic stability in some chaotic dynamical systems \jour Monatsh. Math. \yr 1982 \vol 94 \pages 313--333 \endref \ref \no 11 \bysame \paper On the rate of convergence to equilibrium in one-dimensional systems \jour Comm. Math. Phys. \vol 96 \pages 181--193 \yr 1984 \endref \ref \no 12 \by Y. Kifer \paper On small random perturbations of some smooth dynamical systems \jour Math. USSR-Izv. \vol 8 \yr 1974 \pages 1083--1107 \endref \ref \no 13 \bysame \book Ergodic Theory of Random Transformations \publ Birkh\"auser \publaddr Boston, Basel \yr 1986 \endref \ref \no 14 \bysame \book Random Perturbations of Dynamical Systems \publ Birkh\"auser \publaddr Boston, Basel \yr 1988a \endref \ref \no 15 \bysame \paper A note on integrability of $\CC^r$ norms of stochastic flows and applications \inbook Stochastic Mechanics and Stochastic Processes, Proc. Conf. Swansea/UK 1986 \publ Springer Verlag (Lecture Notes in Math. 1325) \publaddr Berlin \yr 1988b \pages 125--131 \endref \ref \no 16 \by H. Kunita \book Stochastic Flows and Stochastic Differential Equations \publ Cambridge University Press \publaddr Cambridge \yr 1990 \endref \ref \no 17 \by R. Ma\~n\'e \book Ergodic Theory and Differentiable Dynamics \publ Springer-Verlag \yr 1987 \publaddr Berlin Heidelberg New York \endref \ref \no 18 \by D. Mayer \paper On a $\zeta$ function related to the continued fraction transformation \jour Bull. Soc. Math. France \vol 104 \pages 195--203 \yr 1976 \endref \ref \no 19 \by M. Pollicott \paperinfo Private communication (1991) \endref \ref \no 20 \by D. Ruelle \paper Zeta functions for expanding maps and Anosov flows \jour Invent. Math. \yr 1976 \vol 34 \pages 231--242 \endref \ref \no 21 \bysame \paper Locating resonances for Axiom A dynamical systems \jour J. Stat. Phys. \yr 1986 \vol 44 \pages 281--292 \endref \ref \no 22 \bysame \paper The thermodynamic formalism for expanding maps \jour Comm. Math. Phys. \vol 125 \yr 1989 \pages 239--262 \endref \ref \no 23 \bysame \paper An extension of the theory of Fredholm determinants \jour Inst. Hautes \'Etudes Sci. Publ. Math. \vol 72 \pages 175--193 \yr 1990 \endref \ref \no 24 \by M. Rychlik \paper Bounded variation and invariant measures \jour Studia Math. \vol LXXVI \pages 69--80 \yr 1983 \endref \ref \no 25 \by J.H. Wilkinson \book The Algebraic Eigenvalue Problem \yr 1965 \publ Oxford University Press \publaddr London \endref \ref \no 26 \by S. Wong \paper Some metric properties of piecewise monotonic mappings of the unit interval \jour Trans. Amer. Math. Soc. \yr 1978 \pages 493--500 \endref \ref \no 27 \by K. Yosida \book Functional Analysis (Sixth Edition) \publ Springer-Verlag (Grundlehren der mathematischen Wissenschaften 123) \publaddr Berlin Heidelberg New York \yr 1980 \endref \endRefs \enddocument %!PS-Adobe-2.0 %%Creator: MATLAB, The MathWorks, Inc %%Title: MATLAB graph %%CreationDate: 06/14/92 12:12:27 %%Pages: 001 %%BoundingBox: 080 582 532 756 %%DocumentFonts: Times-Roman %%DocumentNeededFonts: Times-Roman %%EndComments % MathWorks dictionary /mathworks 50 dict begin % definition operators /bdef {bind def} bind def /xdef {exch def} bdef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot {gsave} bdef /eplot {grestore} bdef % bounding box in default coordinates /dx 0 def /dy 0 def /sides {/dx urx llx sub def /dy ury lly sub def} bdef /llx 0 def /lly 0 def /urx 0 def /ury 0 def /bbox {/ury xdef /urx xdef /lly xdef /llx xdef sides} bdef % orientation switch /por true def /portrait {/por true def} bdef /landscape {/por false def} bdef % coordinate system mappings /px 8.5 72 mul def /py 11.0 72 mul def /port {dx py div dy px div scale} bdef /land {-90.0 rotate dy neg 0 translate dy py div dx px div scale} bdef /csm {llx lly translate por {port} {land} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [0 4] 0 setdash } bdef /DA { [4] 0 setdash } bdef /DD { [0 4 3 4] 0 setdash } bdef % macros for moveto and polyline /M {moveto} bdef /L {{lineto} repeat stroke} bdef % font control /font_spec () def /lfont currentfont def /sfont currentfont def /selfont {/font_spec xdef} bdef /savefont {font_spec findfont exch scalefont def} bdef /LF {lfont setfont} bdef /SF {sfont setfont} bdef % text display /sh {show} bdef /csh {dup stringwidth pop 2 div neg 0 rmoveto show} bdef /rsh {dup stringwidth pop neg 0 rmoveto show} bdef /r90sh {gsave currentpoint translate 90 rotate csh grestore} bdef currentdict end def %dictionary %%EndProlog %%BeginSetup mathworks begin % fonts for text, standard numbers and exponents %%IncludeFont: Times-Roman /Times-Roman selfont /lfont 20 savefont /sfont 14 savefont %line width, line cap, and joint spec .5 setlinewidth 1 setlinecap 1 setlinejoin end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 080 366 532 756 mathworks begin bpage %%EndPageSetup %%BeginObject: graph 1 bplot 80 582 306 756 bbox portrait csm SO 170.88 77.33 664.22 77.33 664.22 570.67 170.88 570.67 170.88 77.33 M 4 L LF 165.88 71.33 M (0) rsh 170.88 126.66 177.61 126.66 M 1 L 657.48 126.66 664.22 126.66 M 1 L 165.88 120.66 M (0.1) rsh 170.88 176.00 177.61 176.00 M 1 L 657.48 176.00 664.22 176.00 M 1 L 165.88 170.00 M (0.2) rsh 170.88 225.33 177.61 225.33 M 1 L 657.48 225.33 664.22 225.33 M 1 L 165.88 219.33 M (0.3) rsh 170.88 274.67 177.61 274.67 M 1 L 657.48 274.67 664.22 274.67 M 1 L 165.88 268.67 M (0.4) rsh 170.88 324.00 177.61 324.00 M 1 L 657.48 324.00 664.22 324.00 M 1 L 165.88 318.00 M (0.5) rsh 170.88 373.33 177.61 373.33 M 1 L 657.48 373.33 664.22 373.33 M 1 L 165.88 367.33 M (0.6) rsh 170.88 422.67 177.61 422.67 M 1 L 657.48 422.67 664.22 422.67 M 1 L 165.88 416.67 M (0.7) rsh 170.88 472.00 177.61 472.00 M 1 L 657.48 472.00 664.22 472.00 M 1 L 165.88 466.00 M (0.8) rsh 170.88 521.34 177.61 521.34 M 1 L 657.48 521.34 664.22 521.34 M 1 L 165.88 515.34 M (0.9) rsh 165.88 564.67 M (1) rsh 170.88 55.33 M (0) csh 269.55 77.33 269.55 82.53 M 1 L 269.55 565.47 269.55 570.67 M 1 L 269.55 55.33 M (0.2) csh 368.21 77.33 368.21 82.53 M 1 L 368.21 565.47 368.21 570.67 M 1 L 368.21 55.33 M (0.4) csh 466.88 77.33 466.88 82.53 M 1 L 466.88 565.47 466.88 570.67 M 1 L 466.88 55.33 M (0.6) csh 565.55 77.33 565.55 82.53 M 1 L 565.55 565.47 565.55 570.67 M 1 L 565.55 55.33 M (0.8) csh 664.22 55.33 M (1) csh 170.88 77.33 171.86 78.32 172.85 79.30 173.84 80.29 174.82 81.28 175.81 82.26 176.80 83.25 177.78 84.24 178.77 85.22 179.76 86.21 180.74 87.20 181.73 88.18 182.72 89.17 183.70 90.16 184.69 91.14 185.68 92.13 186.66 93.12 187.65 94.10 188.64 95.09 189.62 96.08 190.61 97.06 191.60 98.05 192.58 99.04 193.57 100.02 194.56 101.01 195.54 102.00 196.53 102.98 197.52 103.97 198.50 104.96 199.49 105.94 200.48 106.93 201.46 107.92 202.45 108.90 203.44 109.89 204.42 110.88 205.41 111.86 206.40 112.85 207.38 113.84 208.37 114.82 209.36 115.81 210.34 116.80 211.33 117.78 212.32 118.77 213.30 119.76 214.29 120.74 215.28 121.73 216.26 122.72 217.25 123.70 218.24 124.69 219.22 125.68 220.21 126.66 221.20 127.65 222.18 128.64 223.17 129.62 224.16 130.61 225.14 131.60 226.13 132.58 227.12 133.57 228.11 134.56 229.09 135.54 230.08 136.53 231.07 137.52 232.05 138.50 233.04 139.49 234.03 140.48 235.01 141.46 236.00 142.45 236.99 143.44 237.97 144.42 238.96 145.41 239.95 146.40 240.93 147.39 241.92 148.37 242.91 149.36 243.89 150.35 244.88 151.33 245.87 152.32 246.85 153.31 247.84 154.29 248.83 155.28 249.81 156.27 250.80 157.25 251.79 158.24 252.77 159.23 253.76 160.21 254.75 161.20 255.73 162.19 256.72 163.17 257.71 164.16 258.69 165.15 259.68 166.13 260.67 167.12 261.65 168.11 262.64 169.09 263.63 170.08 264.61 171.07 265.60 172.05 266.59 173.04 267.57 174.03 268.56 175.01 M 99 L 268.56 175.01 269.55 176.00 270.53 176.99 271.52 177.97 272.51 178.96 273.49 179.95 274.48 180.93 275.47 181.92 276.45 182.91 277.44 183.89 278.43 184.88 279.41 185.87 280.40 186.85 281.39 187.84 282.37 188.83 283.36 189.81 284.35 190.80 285.33 191.79 286.32 192.77 287.31 193.76 288.29 194.75 289.28 195.73 290.27 196.72 291.25 197.71 292.24 198.69 293.23 199.68 294.21 200.67 295.20 201.65 296.19 202.64 297.17 203.63 298.16 204.61 299.15 205.60 300.13 206.59 301.12 207.57 302.11 208.56 303.09 209.55 304.08 210.53 305.07 211.52 306.05 212.51 307.04 213.49 308.03 214.48 309.01 215.47 310.00 216.45 310.99 217.44 311.97 218.43 312.96 219.41 313.95 220.40 314.93 221.39 315.92 222.37 316.91 223.36 317.89 224.35 318.88 225.33 319.87 226.32 320.85 227.31 321.84 228.29 322.83 229.28 323.81 230.27 324.80 231.25 325.79 232.24 326.77 233.23 327.76 234.21 328.75 235.20 329.73 236.19 330.72 237.17 331.71 238.16 332.69 239.15 333.68 240.13 334.67 241.12 335.65 242.11 336.64 243.09 337.63 244.08 338.61 245.07 339.60 246.05 340.59 247.04 341.57 248.03 342.56 249.01 343.55 250.00 344.53 250.99 345.52 251.97 346.51 252.96 347.49 253.95 348.48 254.93 349.47 255.92 350.45 256.91 351.44 257.89 352.43 258.88 353.41 259.87 354.40 260.85 355.39 261.84 356.37 262.83 357.36 263.81 358.35 264.80 359.33 265.79 360.32 266.77 361.31 267.76 362.29 268.75 363.28 269.73 364.27 270.72 365.25 271.71 366.24 272.69 M 99 L 366.24 272.69 367.23 273.68 368.21 274.67 369.20 275.65 370.19 276.64 371.17 277.63 372.16 278.61 373.15 279.60 374.13 280.59 375.12 281.57 376.11 282.56 377.09 283.55 378.08 284.53 379.07 285.52 380.05 286.51 381.04 287.49 382.03 288.48 383.01 289.47 384.00 290.45 384.99 291.44 385.97 292.43 386.96 293.41 387.95 294.40 388.93 295.39 389.92 296.37 390.91 297.36 391.89 298.35 392.88 299.33 393.87 300.32 394.85 301.31 395.84 302.29 396.83 303.28 397.81 304.27 398.80 305.25 399.79 306.24 400.77 307.23 401.76 308.21 402.75 309.20 403.73 310.19 404.72 311.17 405.71 312.16 406.69 313.15 407.68 314.13 408.67 315.12 409.65 316.11 410.64 317.09 411.63 318.08 412.61 319.07 413.60 320.05 414.59 321.04 415.57 322.03 416.56 323.01 417.55 324.00 418.53 324.99 419.52 325.97 420.51 326.96 421.49 327.95 422.48 328.93 423.47 329.92 424.45 330.91 425.44 331.89 426.43 332.88 427.41 333.87 428.40 334.85 429.39 335.84 430.37 336.83 431.36 337.81 432.35 338.80 433.33 339.79 434.32 340.77 435.31 341.76 436.29 342.75 437.28 343.73 438.27 344.72 439.25 345.71 440.24 346.69 441.23 347.68 442.21 348.67 443.20 349.65 444.19 350.64 445.17 351.63 446.16 352.61 447.15 353.60 448.13 354.59 449.12 355.57 450.11 356.56 451.09 357.55 452.08 358.53 453.07 359.52 454.05 360.51 455.04 361.49 456.03 362.48 457.01 363.47 458.00 364.45 458.99 365.44 459.97 366.43 460.96 367.41 461.95 368.40 462.93 369.39 463.92 370.37 M 99 L 463.92 370.37 464.91 371.36 465.89 372.35 466.88 373.33 467.87 374.32 468.85 375.31 469.84 376.29 470.83 377.28 471.81 378.27 472.80 379.25 473.79 380.24 474.77 381.23 475.76 382.21 476.75 383.20 477.73 384.19 478.72 385.17 479.71 386.16 480.69 387.15 481.68 388.13 482.67 389.12 483.65 390.11 484.64 391.09 485.63 392.08 486.61 393.07 487.60 394.05 488.59 395.04 489.57 396.03 490.56 397.01 491.55 398.00 492.53 398.99 493.52 399.97 494.51 400.96 495.49 401.95 496.48 402.93 497.47 403.92 498.45 404.91 499.44 405.89 500.43 406.88 501.41 407.87 502.40 408.85 503.39 409.84 504.37 410.83 505.36 411.81 506.35 412.80 507.33 413.79 508.32 414.77 509.31 415.76 510.29 416.75 511.28 417.73 512.27 418.72 513.25 419.71 514.24 420.69 515.23 421.68 516.21 422.67 517.20 423.65 518.19 424.64 519.17 425.63 520.16 426.61 521.15 427.60 522.13 428.59 523.12 429.57 524.11 430.56 525.09 431.55 526.08 432.53 527.07 433.52 528.05 434.51 529.04 435.49 530.03 436.48 531.01 437.47 532.00 438.45 532.99 439.44 533.97 440.43 534.96 441.41 535.95 442.40 536.93 443.39 537.92 444.37 538.91 445.36 539.89 446.35 540.88 447.33 541.87 448.32 542.85 449.31 543.84 450.29 544.83 451.28 545.81 452.27 546.80 453.25 547.79 454.24 548.77 455.23 549.76 456.21 550.75 457.20 551.73 458.19 552.72 459.17 553.71 460.16 554.69 461.15 555.68 462.13 556.67 463.12 557.65 464.11 558.64 465.09 559.63 466.08 560.61 467.07 561.60 468.05 M 99 L 561.60 468.05 562.59 469.04 563.57 470.03 564.56 471.01 565.55 472.00 566.53 472.99 567.52 473.97 568.51 474.96 569.49 475.95 570.48 476.93 571.47 477.92 572.45 478.91 573.44 479.89 574.43 480.88 575.41 481.87 576.40 482.85 577.39 483.84 578.37 484.83 579.36 485.81 580.35 486.80 581.33 487.79 582.32 488.77 583.31 489.76 584.30 490.75 585.28 491.73 586.27 492.72 587.26 493.71 588.24 494.69 589.23 495.68 590.22 496.67 591.20 497.65 592.19 498.64 593.18 499.63 594.16 500.61 595.15 501.60 596.14 502.59 597.12 503.58 598.11 504.56 599.10 505.55 600.08 506.54 601.07 507.52 602.06 508.51 603.04 509.50 604.03 510.48 605.02 511.47 606.00 512.46 606.99 513.44 607.98 514.43 608.96 515.42 609.95 516.40 610.94 517.39 611.92 518.38 612.91 519.36 613.90 520.35 614.88 521.34 615.87 522.32 616.86 523.31 617.84 524.30 618.83 525.28 619.82 526.27 620.80 527.26 621.79 528.24 622.78 529.23 623.76 530.22 624.75 531.20 625.74 532.19 626.72 533.18 627.71 534.16 628.70 535.15 629.68 536.14 630.67 537.12 631.66 538.11 632.64 539.10 633.63 540.08 634.62 541.07 635.60 542.06 636.59 543.04 637.58 544.03 638.56 545.02 639.55 546.00 640.54 546.99 641.52 547.98 642.51 548.96 643.50 549.95 644.48 550.94 645.47 551.92 646.46 552.91 647.44 553.90 648.43 554.88 649.42 555.87 650.40 556.86 651.39 557.84 652.38 558.83 653.36 559.82 654.35 560.80 655.34 561.79 656.32 562.78 657.31 563.76 658.30 564.75 659.28 565.74 M 99 L 659.28 565.74 660.27 566.72 661.26 567.71 662.24 568.70 663.23 569.68 664.22 570.67 M 5 L 170.88 570.67 171.86 566.72 172.85 562.78 173.84 558.83 174.82 554.88 175.81 550.94 176.80 546.99 177.78 543.04 178.77 539.10 179.76 535.15 180.74 531.20 181.73 527.26 182.72 523.31 183.70 519.36 184.69 515.42 185.68 511.47 186.66 507.52 187.65 503.58 188.64 499.63 189.62 495.68 190.61 491.73 191.60 487.79 192.58 483.84 193.57 479.89 194.56 475.95 195.54 472.00 196.53 468.05 197.52 464.11 198.50 460.16 199.49 456.21 200.48 452.27 201.46 448.32 202.45 444.37 203.44 440.43 204.42 436.48 205.41 432.53 206.40 428.59 207.38 424.64 208.37 420.69 209.36 416.75 210.34 412.80 211.33 408.85 212.32 404.91 213.30 400.96 214.29 397.01 215.28 393.07 216.26 389.12 217.25 385.17 218.24 381.23 219.22 377.28 220.21 373.33 221.20 369.39 222.18 365.44 223.17 361.49 224.16 357.55 225.14 353.60 226.13 349.65 227.12 345.71 228.11 341.76 229.09 337.81 230.08 333.87 231.07 329.92 232.05 325.97 233.04 322.03 234.03 318.08 235.01 314.13 236.00 310.19 236.99 306.24 237.97 302.29 238.96 298.35 239.95 294.40 240.93 290.45 241.92 286.51 242.91 282.56 243.89 278.61 244.88 274.67 245.87 270.72 246.85 266.77 247.84 262.83 248.83 258.88 249.81 254.93 250.80 250.99 251.79 247.04 252.77 243.09 253.76 239.15 254.75 235.20 255.73 231.25 256.72 227.31 257.71 223.36 258.69 219.41 259.68 215.47 260.67 211.52 261.65 207.57 262.64 203.63 263.63 199.68 264.61 195.73 265.60 191.79 266.59 187.84 267.57 183.89 268.56 179.95 M 99 L 268.56 179.95 269.55 176.00 270.53 172.05 271.52 168.11 272.51 164.16 273.49 160.21 274.48 156.27 275.47 152.32 276.45 148.37 277.44 144.42 278.43 140.48 279.41 136.53 280.40 132.58 281.39 128.64 282.37 124.69 283.36 120.74 284.35 116.80 285.33 112.85 286.32 108.90 287.31 104.96 288.29 101.01 289.28 97.06 290.27 93.12 291.25 89.17 292.24 85.22 293.23 81.28 294.21 77.33 295.20 79.30 296.19 81.28 297.17 83.25 298.16 85.22 299.15 87.20 300.13 89.17 301.12 91.14 302.11 93.12 303.09 95.09 304.08 97.06 305.07 99.04 306.05 101.01 307.04 102.98 308.03 104.96 309.01 106.93 310.00 108.90 310.99 110.88 311.97 112.85 312.96 114.82 313.95 116.80 314.93 118.77 315.92 120.74 316.91 122.72 317.89 124.69 318.88 126.66 319.87 128.64 320.85 130.61 321.84 132.58 322.83 134.56 323.81 136.53 324.80 138.50 325.79 140.48 326.77 142.45 327.76 144.42 328.75 146.40 329.73 148.37 330.72 150.35 331.71 152.32 332.69 154.29 333.68 156.27 334.67 158.24 335.65 160.21 336.64 162.19 337.63 164.16 338.61 166.13 339.60 168.11 340.59 170.08 341.57 172.05 342.56 174.03 343.55 176.00 344.53 177.97 345.52 179.95 346.51 181.92 347.49 183.89 348.48 185.87 349.47 187.84 350.45 189.81 351.44 191.79 352.43 193.76 353.41 195.73 354.40 197.71 355.39 199.68 356.37 201.65 357.36 203.63 358.35 205.60 359.33 207.57 360.32 209.55 361.31 211.52 362.29 213.49 363.28 215.47 364.27 217.44 365.25 219.41 366.24 221.39 M 99 L 366.24 221.39 367.23 223.36 368.21 225.33 369.20 227.31 370.19 229.28 371.17 231.25 372.16 233.23 373.15 235.20 374.13 237.17 375.12 239.15 376.11 241.12 377.09 243.09 378.08 245.07 379.07 247.04 380.05 249.01 381.04 250.99 382.03 252.96 383.01 254.93 384.00 256.91 384.99 258.88 385.97 260.85 386.96 262.83 387.95 264.80 388.93 266.77 389.92 268.75 390.91 270.72 391.89 272.69 392.88 274.67 393.87 276.64 394.85 278.61 395.84 280.59 396.83 282.56 397.81 284.53 398.80 286.51 399.79 288.48 400.77 290.45 401.76 292.43 402.75 294.40 403.73 296.37 404.72 298.35 405.71 300.32 406.69 302.29 407.68 304.27 408.67 306.24 409.65 308.21 410.64 310.19 411.63 312.16 412.61 314.13 413.60 316.11 414.59 318.08 415.57 320.05 416.56 322.03 417.55 324.00 418.53 322.03 419.52 320.05 420.51 318.08 421.49 316.11 422.48 314.13 423.47 312.16 424.45 310.19 425.44 308.21 426.43 306.24 427.41 304.27 428.40 302.29 429.39 300.32 430.37 298.35 431.36 296.37 432.35 294.40 433.33 292.43 434.32 290.45 435.31 288.48 436.29 286.51 437.28 284.53 438.27 282.56 439.25 280.59 440.24 278.61 441.23 276.64 442.21 274.67 443.20 272.69 444.19 270.72 445.17 268.75 446.16 266.77 447.15 264.80 448.13 262.83 449.12 260.85 450.11 258.88 451.09 256.91 452.08 254.93 453.07 252.96 454.05 250.99 455.04 249.01 456.03 247.04 457.01 245.07 458.00 243.09 458.99 241.12 459.97 239.15 460.96 237.17 461.95 235.20 462.93 233.23 463.92 231.25 M 99 L 463.92 231.25 464.91 229.28 465.89 227.31 466.88 225.33 467.87 223.36 468.85 221.39 469.84 219.41 470.83 217.44 471.81 215.47 472.80 213.49 473.79 211.52 474.77 209.55 475.76 207.57 476.75 205.60 477.73 203.63 478.72 201.65 479.71 199.68 480.69 197.71 481.68 195.73 482.67 193.76 483.65 191.79 484.64 189.81 485.63 187.84 486.61 185.87 487.60 183.89 488.59 181.92 489.57 179.95 490.56 177.97 491.55 176.00 492.53 174.03 493.52 172.05 494.51 170.08 495.49 168.11 496.48 166.13 497.47 164.16 498.45 162.19 499.44 160.21 500.43 158.24 501.41 156.27 502.40 154.29 503.39 152.32 504.37 150.35 505.36 148.37 506.35 146.40 507.33 144.42 508.32 142.45 509.31 140.48 510.29 138.50 511.28 136.53 512.27 134.56 513.25 132.58 514.24 130.61 515.23 128.64 516.21 126.66 517.20 124.69 518.19 122.72 519.17 120.74 520.16 118.77 521.15 116.80 522.13 114.82 523.12 112.85 524.11 110.88 525.09 108.90 526.08 106.93 527.07 104.96 528.05 102.98 529.04 101.01 530.03 99.04 531.01 97.06 532.00 95.09 532.99 93.12 533.97 91.14 534.96 89.17 535.95 87.20 536.93 85.22 537.92 83.25 538.91 81.28 539.89 79.30 540.88 77.33 541.87 81.28 542.85 85.22 543.84 89.17 544.83 93.12 545.81 97.06 546.80 101.01 547.79 104.96 548.77 108.90 549.76 112.85 550.75 116.80 551.73 120.74 552.72 124.69 553.71 128.64 554.69 132.58 555.68 136.53 556.67 140.48 557.65 144.42 558.64 148.37 559.63 152.32 560.61 156.27 561.60 160.21 M 99 L 561.60 160.21 562.59 164.16 563.57 168.11 564.56 172.05 565.55 176.00 566.53 179.95 567.52 183.89 568.51 187.84 569.49 191.79 570.48 195.73 571.47 199.68 572.45 203.63 573.44 207.57 574.43 211.52 575.41 215.47 576.40 219.41 577.39 223.36 578.37 227.31 579.36 231.25 580.35 235.20 581.33 239.15 582.32 243.09 583.31 247.04 584.30 250.99 585.28 254.93 586.27 258.88 587.26 262.83 588.24 266.77 589.23 270.72 590.22 274.67 591.20 278.61 592.19 282.56 593.18 286.51 594.16 290.45 595.15 294.40 596.14 298.35 597.12 302.29 598.11 306.24 599.10 310.19 600.08 314.13 601.07 318.08 602.06 322.03 603.04 325.97 604.03 329.92 605.02 333.87 606.00 337.81 606.99 341.76 607.98 345.71 608.96 349.65 609.95 353.60 610.94 357.55 611.92 361.49 612.91 365.44 613.90 369.39 614.88 373.33 615.87 377.28 616.86 381.23 617.84 385.17 618.83 389.12 619.82 393.07 620.80 397.01 621.79 400.96 622.78 404.91 623.76 408.85 624.75 412.80 625.74 416.75 626.72 420.69 627.71 424.64 628.70 428.59 629.68 432.53 630.67 436.48 631.66 440.43 632.64 444.37 633.63 448.32 634.62 452.27 635.60 456.21 636.59 460.16 637.58 464.11 638.56 468.05 639.55 472.00 640.54 475.95 641.52 479.89 642.51 483.84 643.50 487.79 644.48 491.73 645.47 495.68 646.46 499.63 647.44 503.58 648.43 507.52 649.42 511.47 650.40 515.42 651.39 519.36 652.38 523.31 653.36 527.26 654.35 531.20 655.34 535.15 656.32 539.10 657.31 543.04 658.30 546.99 659.28 550.94 M 99 L 659.28 550.94 660.27 554.88 661.26 558.83 662.24 562.78 663.23 566.72 664.22 570.67 M 5 L 175.81 570.67 175.81 570.67 176.80 566.72 177.78 562.78 178.77 558.83 179.76 554.88 180.74 550.94 181.73 546.99 182.72 543.04 183.70 539.10 184.69 535.15 185.68 531.20 186.66 527.26 187.65 523.31 188.64 519.36 189.62 515.42 190.61 511.47 191.60 507.52 192.58 503.58 193.57 499.63 194.56 495.68 195.54 491.73 196.53 487.79 197.52 483.84 198.50 479.89 199.49 475.95 200.48 472.00 201.46 468.05 202.45 464.11 203.44 460.16 204.42 456.21 205.41 452.27 206.40 448.32 207.38 444.37 208.37 440.43 209.36 436.48 210.34 432.53 211.33 428.59 212.32 424.64 213.30 420.69 214.29 416.75 215.28 412.80 216.26 408.85 217.25 404.91 218.24 400.96 219.22 397.01 220.21 393.07 221.20 389.12 222.18 385.17 223.17 381.23 224.16 377.28 225.14 373.33 226.13 369.39 227.12 365.44 228.11 361.49 229.09 357.55 230.08 353.60 231.07 349.65 232.05 345.71 233.04 341.76 234.03 337.81 235.01 333.87 236.00 329.92 236.99 325.97 237.97 322.03 238.96 318.08 239.95 314.13 240.93 310.19 241.92 306.24 242.91 302.29 243.89 298.35 244.88 294.40 245.87 290.45 246.85 286.51 247.84 282.56 248.83 278.61 249.81 274.67 250.80 270.72 251.79 266.77 252.77 262.83 253.76 258.88 254.75 254.93 255.73 250.99 256.72 247.04 257.71 243.09 258.69 239.15 259.68 235.20 260.67 231.25 261.65 227.31 262.64 223.36 263.63 219.41 264.61 215.47 265.60 211.52 266.59 207.57 267.57 203.63 268.56 199.68 269.55 195.73 270.53 191.79 271.52 187.84 272.51 183.89 M 99 L 272.51 183.89 273.49 179.95 274.48 176.00 275.47 172.05 276.45 168.11 277.44 164.16 278.43 160.21 279.41 156.27 280.40 152.32 281.39 148.37 282.37 144.42 283.36 140.48 284.35 136.53 285.33 132.58 286.32 128.64 287.31 124.69 288.29 120.74 289.28 116.80 290.27 112.85 291.25 108.90 292.24 104.96 293.23 101.01 294.21 97.06 295.20 99.04 296.19 101.01 297.17 102.98 298.16 104.96 299.15 106.93 300.13 108.90 301.12 110.88 302.11 112.85 303.09 114.82 304.08 116.80 305.07 118.77 306.05 120.74 307.04 122.72 308.03 124.69 309.01 126.66 310.00 128.64 310.99 130.61 311.97 132.58 312.96 134.56 313.95 136.53 314.93 138.50 315.92 140.48 316.91 142.45 317.89 144.42 318.88 146.40 319.87 148.37 320.85 150.35 321.84 152.32 322.83 154.29 323.81 156.27 324.80 158.24 325.79 160.21 326.77 162.19 327.76 164.16 328.75 166.13 329.73 168.11 330.72 170.08 331.71 172.05 332.69 174.03 333.68 176.00 334.67 177.97 335.65 179.95 336.64 181.92 337.63 183.89 338.61 185.87 339.60 187.84 340.59 189.81 341.57 191.79 342.56 193.76 343.55 195.73 344.53 197.71 345.52 199.68 346.51 201.65 347.49 203.63 348.48 205.60 349.47 207.57 350.45 209.55 351.44 211.52 352.43 213.49 353.41 215.47 354.40 217.44 355.39 219.41 356.37 221.39 357.36 223.36 358.35 225.33 359.33 227.31 360.32 229.28 361.31 231.25 362.29 233.23 363.28 235.20 364.27 237.17 365.25 239.15 366.24 241.12 367.23 243.09 368.21 245.07 369.20 247.04 370.19 249.01 M 99 L 370.19 249.01 371.17 250.99 372.16 252.96 373.15 254.93 374.13 256.91 375.12 258.88 376.11 260.85 377.09 262.83 378.08 264.80 379.07 266.77 380.05 268.75 381.04 270.72 382.03 272.69 383.01 274.67 384.00 276.64 384.99 278.61 385.97 280.59 386.96 282.56 387.95 284.53 388.93 286.51 389.92 288.48 390.91 290.45 391.89 292.43 392.88 294.40 393.87 296.37 394.85 298.35 395.84 300.32 396.83 302.29 397.81 304.27 398.80 306.24 399.79 308.21 400.77 310.19 401.76 312.16 402.75 314.13 403.73 316.11 404.72 318.08 405.71 320.05 406.69 322.03 407.68 324.00 408.67 325.97 409.65 327.95 410.64 329.92 411.63 331.89 412.61 333.87 413.60 335.84 414.59 337.81 415.57 339.79 416.56 341.76 417.55 343.73 418.53 341.76 419.52 339.79 420.51 337.81 421.49 335.84 422.48 333.87 423.47 331.89 424.45 329.92 425.44 327.95 426.43 325.97 427.41 324.00 428.40 322.03 429.39 320.05 430.37 318.08 431.36 316.11 432.35 314.13 433.33 312.16 434.32 310.19 435.31 308.21 436.29 306.24 437.28 304.27 438.27 302.29 439.25 300.32 440.24 298.35 441.23 296.37 442.21 294.40 443.20 292.43 444.19 290.45 445.17 288.48 446.16 286.51 447.15 284.53 448.13 282.56 449.12 280.59 450.11 278.61 451.09 276.64 452.08 274.67 453.07 272.69 454.05 270.72 455.04 268.75 456.03 266.77 457.01 264.80 458.00 262.83 458.99 260.85 459.97 258.88 460.96 256.91 461.95 254.93 462.93 252.96 463.92 250.99 464.91 249.01 465.89 247.04 466.88 245.07 467.87 243.09 M 99 L 467.87 243.09 468.85 241.12 469.84 239.15 470.83 237.17 471.81 235.20 472.80 233.23 473.79 231.25 474.77 229.28 475.76 227.31 476.75 225.33 477.73 223.36 478.72 221.39 479.71 219.41 480.69 217.44 481.68 215.47 482.67 213.49 483.65 211.52 484.64 209.55 485.63 207.57 486.61 205.60 487.60 203.63 488.59 201.65 489.57 199.68 490.56 197.71 491.55 195.73 492.53 193.76 493.52 191.79 494.51 189.81 495.49 187.84 496.48 185.87 497.47 183.89 498.45 181.92 499.44 179.95 500.43 177.97 501.41 176.00 502.40 174.03 503.39 172.05 504.37 170.08 505.36 168.11 506.35 166.13 507.33 164.16 508.32 162.19 509.31 160.21 510.29 158.24 511.28 156.27 512.27 154.29 513.25 152.32 514.24 150.35 515.23 148.37 516.21 146.40 517.20 144.42 518.19 142.45 519.17 140.48 520.16 138.50 521.15 136.53 522.13 134.56 523.12 132.58 524.11 130.61 525.09 128.64 526.08 126.66 527.07 124.69 528.05 122.72 529.04 120.74 530.03 118.77 531.01 116.80 532.00 114.82 532.99 112.85 533.97 110.88 534.96 108.90 535.95 106.93 536.93 104.96 537.92 102.98 538.91 101.01 539.89 99.04 540.88 97.06 541.87 101.01 542.85 104.96 543.84 108.90 544.83 112.85 545.81 116.80 546.80 120.74 547.79 124.69 548.77 128.64 549.76 132.58 550.75 136.53 551.73 140.48 552.72 144.42 553.71 148.37 554.69 152.32 555.68 156.27 556.67 160.21 557.65 164.16 558.64 168.11 559.63 172.05 560.61 176.00 561.60 179.95 562.59 183.89 563.57 187.84 564.56 191.79 565.55 195.73 M 99 L 565.55 195.73 566.53 199.68 567.52 203.63 568.51 207.57 569.49 211.52 570.48 215.47 571.47 219.41 572.45 223.36 573.44 227.31 574.43 231.25 575.41 235.20 576.40 239.15 577.39 243.09 578.37 247.04 579.36 250.99 580.35 254.93 581.33 258.88 582.32 262.83 583.31 266.77 584.30 270.72 585.28 274.67 586.27 278.61 587.26 282.56 588.24 286.51 589.23 290.45 590.22 294.40 591.20 298.35 592.19 302.29 593.18 306.24 594.16 310.19 595.15 314.13 596.14 318.08 597.12 322.03 598.11 325.97 599.10 329.92 600.08 333.87 601.07 337.81 602.06 341.76 603.04 345.71 604.03 349.65 605.02 353.60 606.00 357.55 606.99 361.49 607.98 365.44 608.96 369.39 609.95 373.33 610.94 377.28 611.92 381.23 612.91 385.17 613.90 389.12 614.88 393.07 615.87 397.01 616.86 400.96 617.84 404.91 618.83 408.85 619.82 412.80 620.80 416.75 621.79 420.69 622.78 424.64 623.76 428.59 624.75 432.53 625.74 436.48 626.72 440.43 627.71 444.37 628.70 448.32 629.68 452.27 630.67 456.21 631.66 460.16 632.64 464.11 633.63 468.05 634.62 472.00 635.60 475.95 636.59 479.89 637.58 483.84 638.56 487.79 639.55 491.73 640.54 495.68 641.52 499.63 642.51 503.58 643.50 507.52 644.48 511.47 645.47 515.42 646.46 519.36 647.44 523.31 648.43 527.26 649.42 531.20 650.40 535.15 651.39 539.10 652.38 543.04 653.36 546.99 654.35 550.94 655.34 554.88 656.32 558.83 657.31 562.78 658.30 566.72 659.28 570.67 659.28 570.67 M 96 L 417.55 586.00 M (1 iterate, epsilon= 0.04) csh eplot %%EndObject graph 1 %%BeginObject: graph 2 bplot 306 582 532 756 bbox portrait csm SO 170.88 77.33 664.22 77.33 664.22 570.67 170.88 570.67 170.88 77.33 M 4 L LF 165.88 71.33 M (0) rsh 170.88 126.66 177.61 126.66 M 1 L 657.48 126.66 664.22 126.66 M 1 L 165.88 120.66 M (0.1) rsh 170.88 176.00 177.61 176.00 M 1 L 657.48 176.00 664.22 176.00 M 1 L 165.88 170.00 M (0.2) rsh 170.88 225.33 177.61 225.33 M 1 L 657.48 225.33 664.22 225.33 M 1 L 165.88 219.33 M (0.3) rsh 170.88 274.67 177.61 274.67 M 1 L 657.48 274.67 664.22 274.67 M 1 L 165.88 268.67 M (0.4) rsh 170.88 324.00 177.61 324.00 M 1 L 657.48 324.00 664.22 324.00 M 1 L 165.88 318.00 M (0.5) rsh 170.88 373.33 177.61 373.33 M 1 L 657.48 373.33 664.22 373.33 M 1 L 165.88 367.33 M (0.6) rsh 170.88 422.67 177.61 422.67 M 1 L 657.48 422.67 664.22 422.67 M 1 L 165.88 416.67 M (0.7) rsh 170.88 472.00 177.61 472.00 M 1 L 657.48 472.00 664.22 472.00 M 1 L 165.88 466.00 M (0.8) rsh 170.88 521.34 177.61 521.34 M 1 L 657.48 521.34 664.22 521.34 M 1 L 165.88 515.34 M (0.9) rsh 165.88 564.67 M (1) rsh 170.88 55.33 M (0.4) csh 294.21 77.33 294.21 82.53 M 1 L 294.21 565.47 294.21 570.67 M 1 L 294.21 55.33 M (0.45) csh 417.55 77.33 417.55 82.53 M 1 L 417.55 565.47 417.55 570.67 M 1 L 417.55 55.33 M (0.5) csh 540.88 77.33 540.88 82.53 M 1 L 540.88 565.47 540.88 570.67 M 1 L 540.88 55.33 M (0.55) csh 664.22 55.33 M (0.6) csh 170.88 274.67 170.88 274.67 172.11 274.91 173.34 275.16 174.58 275.41 175.81 275.65 177.04 275.90 178.28 276.15 179.51 276.39 180.74 276.64 181.98 276.89 183.21 277.13 184.44 277.38 185.68 277.63 186.91 277.87 188.14 278.12 189.38 278.37 190.61 278.61 191.84 278.86 193.08 279.11 194.31 279.35 195.54 279.60 196.78 279.85 198.01 280.09 199.24 280.34 200.48 280.59 201.71 280.83 202.94 281.08 204.18 281.33 205.41 281.57 206.64 281.82 207.88 282.07 209.11 282.31 210.34 282.56 211.58 282.81 212.81 283.05 214.04 283.30 215.28 283.55 216.51 283.79 217.74 284.04 218.98 284.29 220.21 284.53 221.44 284.78 222.68 285.03 223.91 285.27 225.14 285.52 226.38 285.77 227.61 286.01 228.85 286.26 230.08 286.51 231.31 286.75 232.55 287.00 233.78 287.25 235.01 287.49 236.25 287.74 237.48 287.99 238.71 288.23 239.95 288.48 241.18 288.73 242.41 288.97 243.65 289.22 244.88 289.47 246.11 289.71 247.35 289.96 248.58 290.21 249.81 290.45 251.05 290.70 252.28 290.95 253.51 291.19 254.75 291.44 255.98 291.69 257.21 291.93 258.45 292.18 259.68 292.43 260.91 292.67 262.15 292.92 263.38 293.17 264.61 293.41 265.85 293.66 267.08 293.91 268.31 294.15 269.55 294.40 270.78 294.65 272.01 294.89 273.25 295.14 274.48 295.39 275.71 295.63 276.95 295.88 278.18 296.13 279.41 296.37 280.65 296.62 281.88 296.87 283.11 297.11 284.35 297.36 285.58 297.61 286.81 297.85 288.05 298.10 289.28 298.35 290.51 298.59 291.75 298.84 M 99 L 291.75 298.84 292.98 299.09 294.21 299.33 295.45 299.58 296.68 299.83 297.91 300.07 299.15 300.32 300.38 300.57 301.61 300.81 302.85 301.06 304.08 301.31 305.31 301.55 306.55 301.80 307.78 302.05 309.01 302.29 310.25 302.54 311.48 302.79 312.71 303.03 313.95 303.28 315.18 303.53 316.41 303.77 317.65 304.02 318.88 304.27 320.11 304.51 321.35 304.76 322.58 305.01 323.81 305.25 325.05 305.50 326.28 305.75 327.51 305.99 328.75 306.24 329.98 306.49 331.21 306.73 332.45 306.98 333.68 307.23 334.91 307.47 336.15 307.72 337.38 307.97 338.61 308.21 339.85 308.46 341.08 308.71 342.31 308.95 343.55 309.20 344.78 309.45 346.01 309.69 347.25 309.94 348.48 310.19 349.71 310.43 350.95 310.68 352.18 310.93 353.41 311.17 354.65 311.42 355.88 311.67 357.11 311.91 358.35 312.16 359.58 312.41 360.81 312.65 362.05 312.90 363.28 313.15 364.51 313.39 365.75 313.64 366.98 313.89 368.21 314.13 369.45 314.38 370.68 314.63 371.91 314.87 373.15 315.12 374.38 315.37 375.61 315.61 376.85 315.86 378.08 316.11 379.31 316.35 380.55 316.60 381.78 316.85 383.01 317.09 384.25 317.34 385.48 317.59 386.71 317.83 387.95 318.08 389.18 318.33 390.41 318.57 391.65 318.82 392.88 319.07 394.11 319.31 395.35 319.56 396.58 319.81 397.81 320.05 399.05 320.30 400.28 320.55 401.51 320.79 402.75 321.04 403.98 321.29 405.21 321.53 406.45 321.78 407.68 322.03 408.91 322.27 410.15 322.52 411.38 322.77 412.61 323.01 413.85 323.26 M 99 L 413.85 323.26 415.08 323.51 416.31 323.75 417.55 324.00 418.78 324.25 420.01 324.49 421.25 324.74 422.48 324.99 423.71 325.23 424.95 325.48 426.18 325.73 427.41 325.97 428.65 326.22 429.88 326.47 431.11 326.71 432.35 326.96 433.58 327.21 434.81 327.45 436.05 327.70 437.28 327.95 438.51 328.19 439.75 328.44 440.98 328.69 442.21 328.93 443.45 329.18 444.68 329.43 445.91 329.67 447.15 329.92 448.38 330.17 449.61 330.41 450.85 330.66 452.08 330.91 453.31 331.15 454.55 331.40 455.78 331.65 457.01 331.89 458.25 332.14 459.48 332.39 460.71 332.63 461.95 332.88 463.18 333.13 464.41 333.37 465.65 333.62 466.88 333.87 468.11 334.11 469.35 334.36 470.58 334.61 471.81 334.85 473.05 335.10 474.28 335.35 475.51 335.59 476.75 335.84 477.98 336.09 479.21 336.33 480.45 336.58 481.68 336.83 482.91 337.07 484.15 337.32 485.38 337.57 486.61 337.81 487.85 338.06 489.08 338.31 490.31 338.55 491.55 338.80 492.78 339.05 494.01 339.29 495.25 339.54 496.48 339.79 497.71 340.03 498.95 340.28 500.18 340.53 501.41 340.77 502.65 341.02 503.88 341.27 505.11 341.51 506.35 341.76 507.58 342.01 508.81 342.25 510.05 342.50 511.28 342.75 512.51 342.99 513.75 343.24 514.98 343.49 516.21 343.73 517.45 343.98 518.68 344.23 519.91 344.47 521.15 344.72 522.38 344.97 523.61 345.21 524.85 345.46 526.08 345.71 527.31 345.95 528.55 346.20 529.78 346.45 531.01 346.69 532.25 346.94 533.48 347.19 534.71 347.43 535.95 347.68 M 99 L 535.95 347.68 537.18 347.93 538.41 348.17 539.65 348.42 540.88 348.67 542.11 348.91 543.35 349.16 544.58 349.41 545.81 349.65 547.05 349.90 548.28 350.15 549.51 350.39 550.75 350.64 551.98 350.89 553.21 351.13 554.45 351.38 555.68 351.63 556.91 351.87 558.15 352.12 559.38 352.37 560.61 352.61 561.85 352.86 563.08 353.11 564.31 353.35 565.55 353.60 566.78 353.85 568.01 354.09 569.25 354.34 570.48 354.59 571.71 354.83 572.95 355.08 574.18 355.33 575.41 355.57 576.65 355.82 577.88 356.07 579.11 356.31 580.35 356.56 581.58 356.81 582.82 357.05 584.05 357.30 585.28 357.55 586.52 357.79 587.75 358.04 588.98 358.29 590.22 358.53 591.45 358.78 592.68 359.03 593.92 359.27 595.15 359.52 596.38 359.77 597.62 360.01 598.85 360.26 600.08 360.51 601.32 360.75 602.55 361.00 603.78 361.25 605.02 361.49 606.25 361.74 607.48 361.99 608.72 362.23 609.95 362.48 611.18 362.73 612.42 362.97 613.65 363.22 614.88 363.47 616.12 363.71 617.35 363.96 618.58 364.21 619.82 364.45 621.05 364.70 622.28 364.95 623.52 365.19 624.75 365.44 625.98 365.69 627.22 365.93 628.45 366.18 629.68 366.43 630.92 366.67 632.15 366.92 633.38 367.17 634.62 367.41 635.85 367.66 637.08 367.91 638.32 368.15 639.55 368.40 640.78 368.65 642.02 368.89 643.25 369.14 644.48 369.39 645.72 369.63 646.95 369.88 648.18 370.13 649.42 370.37 650.65 370.62 651.88 370.87 653.12 371.11 654.35 371.36 655.58 371.61 656.82 371.85 658.05 372.10 M 99 L 658.05 372.10 659.28 372.35 660.52 372.59 661.75 372.84 662.98 373.09 664.22 373.33 M 5 L 170.88 225.33 170.88 225.33 172.11 233.23 173.34 241.12 174.58 249.01 175.81 256.91 177.04 264.80 178.28 272.69 179.51 280.59 180.74 288.48 181.98 296.37 183.21 304.27 184.44 312.16 185.68 320.05 186.91 320.05 188.14 312.16 189.38 304.27 190.61 296.37 191.84 288.48 193.08 280.59 194.31 272.69 195.54 264.80 196.78 256.91 198.01 249.01 199.24 241.12 200.48 233.23 201.71 225.33 202.94 217.44 204.18 209.55 205.41 201.65 206.64 193.76 207.88 185.87 209.11 177.97 210.34 170.08 211.58 162.19 212.81 154.29 214.04 146.40 215.28 138.50 216.51 130.61 217.74 122.72 218.98 114.82 220.21 106.93 221.44 99.04 222.68 91.14 223.91 83.25 225.14 81.28 226.38 97.06 227.61 112.85 228.85 128.64 230.08 144.42 231.31 160.21 232.55 176.00 233.78 191.79 235.01 207.57 236.25 223.36 237.48 239.15 238.71 254.93 239.95 270.72 241.18 286.51 242.41 302.29 243.65 318.08 244.88 333.87 246.11 349.65 247.35 365.44 248.58 381.23 249.81 397.01 251.05 412.80 252.28 428.59 253.51 444.37 254.75 460.16 255.98 475.95 257.21 491.73 258.45 507.52 259.68 523.31 260.91 539.10 262.15 554.88 263.38 570.67 264.61 562.78 265.85 554.88 267.08 546.99 268.31 539.10 269.55 531.20 270.78 523.31 272.01 515.42 273.25 507.52 274.48 499.63 275.71 491.73 276.95 483.84 278.18 475.95 279.41 468.05 280.65 460.16 281.88 452.27 283.11 444.37 284.35 436.48 285.58 428.59 286.81 420.69 288.05 412.80 289.28 404.91 290.51 397.01 291.75 389.12 M 99 L 291.75 389.12 292.98 381.23 294.21 373.33 295.45 365.44 296.68 357.55 297.91 349.65 299.15 341.76 300.38 333.87 301.61 325.97 302.85 318.08 304.08 310.19 305.31 302.29 306.55 294.40 307.78 286.51 309.01 278.61 310.25 270.72 311.48 262.83 312.71 254.93 313.95 247.04 315.18 239.15 316.41 231.25 317.65 223.36 318.88 215.47 320.11 207.57 321.35 199.68 322.58 191.79 323.81 183.89 325.05 176.00 326.28 168.11 327.51 160.21 328.75 152.32 329.98 144.42 331.21 136.53 332.45 128.64 333.68 120.74 334.91 112.85 336.15 104.96 337.38 97.06 338.61 89.17 339.85 81.28 341.08 79.30 342.31 83.25 343.55 87.20 344.78 91.14 346.01 95.09 347.25 99.04 348.48 102.98 349.71 106.93 350.95 110.88 352.18 114.82 353.41 118.77 354.65 122.72 355.88 126.66 357.11 130.61 358.35 134.56 359.58 138.50 360.81 142.45 362.05 146.40 363.28 150.35 364.51 154.29 365.75 158.24 366.98 162.19 368.21 166.13 369.45 170.08 370.68 174.03 371.91 177.97 373.15 181.92 374.38 185.87 375.61 189.81 376.85 193.76 378.08 197.71 379.31 201.65 380.55 205.60 381.78 209.55 383.01 213.49 384.25 217.44 385.48 221.39 386.71 225.33 387.95 229.28 389.18 233.23 390.41 237.17 391.65 241.12 392.88 245.07 394.11 249.01 395.35 252.96 396.58 256.91 397.81 260.85 399.05 264.80 400.28 268.75 401.51 272.69 402.75 276.64 403.98 280.59 405.21 284.53 406.45 288.48 407.68 292.43 408.91 296.37 410.15 300.32 411.38 304.27 412.61 308.21 413.85 312.16 M 99 L 413.85 312.16 415.08 316.11 416.31 320.05 417.55 324.00 418.78 320.05 420.01 316.11 421.25 312.16 422.48 308.21 423.71 304.27 424.95 300.32 426.18 296.37 427.41 292.43 428.65 288.48 429.88 284.53 431.11 280.59 432.35 276.64 433.58 272.69 434.81 268.75 436.05 264.80 437.28 260.85 438.51 256.91 439.75 252.96 440.98 249.01 442.21 245.07 443.45 241.12 444.68 237.17 445.91 233.23 447.15 229.28 448.38 225.33 449.61 221.39 450.85 217.44 452.08 213.49 453.31 209.55 454.55 205.60 455.78 201.65 457.01 197.71 458.25 193.76 459.48 189.81 460.71 185.87 461.95 181.92 463.18 177.97 464.41 174.03 465.65 170.08 466.88 166.13 468.11 162.19 469.35 158.24 470.58 154.29 471.81 150.35 473.05 146.40 474.28 142.45 475.51 138.50 476.75 134.56 477.98 130.61 479.21 126.66 480.45 122.72 481.68 118.77 482.91 114.82 484.15 110.88 485.38 106.93 486.61 102.98 487.85 99.04 489.08 95.09 490.31 91.14 491.55 87.20 492.78 83.25 494.01 79.30 495.25 81.28 496.48 89.17 497.71 97.06 498.95 104.96 500.18 112.85 501.41 120.74 502.65 128.64 503.88 136.53 505.11 144.42 506.35 152.32 507.58 160.21 508.81 168.11 510.05 176.00 511.28 183.89 512.51 191.79 513.75 199.68 514.98 207.57 516.21 215.47 517.45 223.36 518.68 231.25 519.91 239.15 521.15 247.04 522.38 254.93 523.61 262.83 524.85 270.72 526.08 278.61 527.31 286.51 528.55 294.40 529.78 302.29 531.01 310.19 532.25 318.08 533.48 325.97 534.71 333.87 535.95 341.76 M 99 L 535.95 341.76 537.18 349.65 538.41 357.55 539.65 365.44 540.88 373.33 542.11 381.23 543.35 389.12 544.58 397.01 545.81 404.91 547.05 412.80 548.28 420.69 549.51 428.59 550.75 436.48 551.98 444.37 553.21 452.27 554.45 460.16 555.68 468.05 556.91 475.95 558.15 483.84 559.38 491.73 560.61 499.63 561.85 507.52 563.08 515.42 564.31 523.31 565.55 531.20 566.78 539.10 568.01 546.99 569.25 554.88 570.48 562.78 571.71 570.67 572.95 554.88 574.18 539.10 575.41 523.31 576.65 507.52 577.88 491.73 579.11 475.95 580.35 460.16 581.58 444.37 582.82 428.59 584.05 412.80 585.28 397.01 586.52 381.23 587.75 365.44 588.98 349.65 590.22 333.87 591.45 318.08 592.68 302.29 593.92 286.51 595.15 270.72 596.38 254.93 597.62 239.15 598.85 223.36 600.08 207.57 601.32 191.79 602.55 176.00 603.78 160.21 605.02 144.42 606.25 128.64 607.48 112.85 608.72 97.06 609.95 81.28 611.18 83.25 612.42 91.14 613.65 99.04 614.88 106.93 616.12 114.82 617.35 122.72 618.58 130.61 619.82 138.50 621.05 146.40 622.28 154.29 623.52 162.19 624.75 170.08 625.98 177.97 627.22 185.87 628.45 193.76 629.68 201.65 630.92 209.55 632.15 217.44 633.38 225.33 634.62 233.23 635.85 241.12 637.08 249.01 638.32 256.91 639.55 264.80 640.78 272.69 642.02 280.59 643.25 288.48 644.48 296.37 645.72 304.27 646.95 312.16 648.18 320.05 649.42 320.05 650.65 312.16 651.88 304.27 653.12 296.37 654.35 288.48 655.58 280.59 656.82 272.69 658.05 264.80 M 99 L 658.05 264.80 659.28 256.91 660.52 249.01 661.75 241.12 662.98 233.23 664.22 225.33 M 5 L 170.88 274.67 170.88 274.67 172.11 290.45 173.34 306.24 174.58 322.03 175.81 337.81 177.04 353.60 178.28 369.39 179.51 385.17 180.74 400.96 181.98 416.75 183.21 432.53 184.44 448.32 185.68 464.11 186.91 479.89 188.14 495.68 189.38 511.47 190.61 503.58 191.84 495.68 193.08 487.79 194.31 479.89 195.54 472.00 196.78 464.11 198.01 456.21 199.24 448.32 200.48 440.43 201.71 432.53 202.94 424.64 204.18 416.75 205.41 408.85 206.64 400.96 207.88 393.07 209.11 385.17 210.34 377.28 211.58 369.39 212.81 361.49 214.04 353.60 215.28 345.71 216.51 337.81 217.74 329.92 218.98 322.03 220.21 314.13 221.44 306.24 222.68 298.35 223.91 290.45 225.14 282.56 226.38 274.67 227.61 266.77 228.85 258.88 230.08 250.99 231.31 243.09 232.55 235.20 233.78 227.31 235.01 219.41 236.25 211.52 237.48 203.63 238.71 195.73 239.95 187.84 241.18 179.95 242.41 172.05 243.65 164.16 244.88 156.27 246.11 148.37 247.35 140.48 248.58 132.58 249.81 124.69 251.05 116.80 252.28 108.90 253.51 101.01 254.75 99.04 255.98 102.98 257.21 106.93 258.45 110.88 259.68 114.82 260.91 118.77 262.15 122.72 263.38 126.66 264.61 130.61 265.85 134.56 267.08 138.50 268.31 142.45 269.55 146.40 270.78 150.35 272.01 154.29 273.25 158.24 274.48 162.19 275.71 166.13 276.95 170.08 278.18 174.03 279.41 177.97 280.65 181.92 281.88 185.87 283.11 189.81 284.35 193.76 285.58 197.71 286.81 201.65 288.05 205.60 289.28 209.55 290.51 213.49 291.75 217.44 M 99 L 291.75 217.44 292.98 221.39 294.21 225.33 295.45 229.28 296.68 233.23 297.91 237.17 299.15 241.12 300.38 245.07 301.61 249.01 302.85 252.96 304.08 256.91 305.31 260.85 306.55 264.80 307.78 268.75 309.01 272.69 310.25 276.64 311.48 280.59 312.71 284.53 313.95 288.48 315.18 292.43 316.41 296.37 317.65 300.32 318.88 304.27 320.11 308.21 321.35 312.16 322.58 316.11 323.81 320.05 325.05 324.00 326.28 327.95 327.51 331.89 328.75 335.84 329.98 339.79 331.21 343.73 332.45 339.79 333.68 335.84 334.91 331.89 336.15 327.95 337.38 324.00 338.61 320.05 339.85 316.11 341.08 312.16 342.31 308.21 343.55 304.27 344.78 308.21 346.01 312.16 347.25 316.11 348.48 320.05 349.71 324.00 350.95 327.95 352.18 331.89 353.41 335.84 354.65 339.79 355.88 343.73 357.11 339.79 358.35 335.84 359.58 331.89 360.81 327.95 362.05 324.00 363.28 320.05 364.51 316.11 365.75 312.16 366.98 308.21 368.21 304.27 369.45 308.21 370.68 312.16 371.91 316.11 373.15 320.05 374.38 324.00 375.61 327.95 376.85 331.89 378.08 335.84 379.31 339.79 380.55 343.73 381.78 339.79 383.01 335.84 384.25 331.89 385.48 327.95 386.71 324.00 387.95 320.05 389.18 316.11 390.41 312.16 391.65 308.21 392.88 304.27 394.11 308.21 395.35 312.16 396.58 316.11 397.81 320.05 399.05 324.00 400.28 327.95 401.51 331.89 402.75 335.84 403.98 339.79 405.21 343.73 406.45 339.79 407.68 335.84 408.91 331.89 410.15 327.95 411.38 324.00 412.61 320.05 413.85 316.11 M 99 L 413.85 316.11 415.08 312.16 416.31 308.21 417.55 304.27 418.78 308.21 420.01 312.16 421.25 316.11 422.48 320.05 423.71 324.00 424.95 327.95 426.18 331.89 427.41 335.84 428.65 339.79 429.88 343.73 431.11 339.79 432.35 335.84 433.58 331.89 434.81 327.95 436.05 324.00 437.28 320.05 438.51 316.11 439.75 312.16 440.98 308.21 442.21 304.27 443.45 308.21 444.68 312.16 445.91 316.11 447.15 320.05 448.38 324.00 449.61 327.95 450.85 331.89 452.08 335.84 453.31 339.79 454.55 343.73 455.78 339.79 457.01 335.84 458.25 331.89 459.48 327.95 460.71 324.00 461.95 320.05 463.18 316.11 464.41 312.16 465.65 308.21 466.88 304.27 468.11 308.21 469.35 312.16 470.58 316.11 471.81 320.05 473.05 324.00 474.28 327.95 475.51 331.89 476.75 335.84 477.98 339.79 479.21 343.73 480.45 339.79 481.68 335.84 482.91 331.89 484.15 327.95 485.38 324.00 486.61 320.05 487.85 316.11 489.08 312.16 490.31 308.21 491.55 304.27 492.78 308.21 494.01 312.16 495.25 316.11 496.48 320.05 497.71 324.00 498.95 327.95 500.18 331.89 501.41 335.84 502.65 339.79 503.88 343.73 505.11 339.79 506.35 335.84 507.58 331.89 508.81 327.95 510.05 324.00 511.28 320.05 512.51 316.11 513.75 312.16 514.98 308.21 516.21 304.27 517.45 300.32 518.68 296.37 519.91 292.43 521.15 288.48 522.38 284.53 523.61 280.59 524.85 276.64 526.08 272.69 527.31 268.75 528.55 264.80 529.78 260.85 531.01 256.91 532.25 252.96 533.48 249.01 534.71 245.07 535.95 241.12 M 99 L 535.95 241.12 537.18 237.17 538.41 233.23 539.65 229.28 540.88 225.33 542.11 221.39 543.35 217.44 544.58 213.49 545.81 209.55 547.05 205.60 548.28 201.65 549.51 197.71 550.75 193.76 551.98 189.81 553.21 185.87 554.45 181.92 555.68 177.97 556.91 174.03 558.15 170.08 559.38 166.13 560.61 162.19 561.85 158.24 563.08 154.29 564.31 150.35 565.55 146.40 566.78 142.45 568.01 138.50 569.25 134.56 570.48 130.61 571.71 126.66 572.95 122.72 574.18 118.77 575.41 114.82 576.65 110.88 577.88 106.93 579.11 102.98 580.35 99.04 581.58 101.01 582.82 108.90 584.05 116.80 585.28 124.69 586.52 132.58 587.75 140.48 588.98 148.37 590.22 156.27 591.45 164.16 592.68 172.05 593.92 179.95 595.15 187.84 596.38 195.73 597.62 203.63 598.85 211.52 600.08 219.41 601.32 227.31 602.55 235.20 603.78 243.09 605.02 250.99 606.25 258.88 607.48 266.77 608.72 274.67 609.95 282.56 611.18 290.45 612.42 298.35 613.65 306.24 614.88 314.13 616.12 322.03 617.35 329.92 618.58 337.81 619.82 345.71 621.05 353.60 622.28 361.49 623.52 369.39 624.75 377.28 625.98 385.17 627.22 393.07 628.45 400.96 629.68 408.85 630.92 416.75 632.15 424.64 633.38 432.53 634.62 440.43 635.85 448.32 637.08 456.21 638.32 464.11 639.55 472.00 640.78 479.89 642.02 487.79 643.25 495.68 644.48 503.58 645.72 511.47 646.95 495.68 648.18 479.89 649.42 464.11 650.65 448.32 651.88 432.53 653.12 416.75 654.35 400.96 655.58 385.17 656.82 369.39 658.05 353.60 M 99 L 658.05 353.60 659.28 337.81 660.52 322.03 661.75 306.24 662.98 290.45 664.22 274.67 M 5 L 417.55 586.00 M (4 iterates, epsilon= 0.04) csh eplot %%EndObject graph 4 epage end showpage %%Trailer ENDBODY