97-149 Gundlach V. M., Latushkin Y.

A formula for the essential spectral radius of Ruelle's transfer operator on smooth H\"older spaces (39K, AMS-LaTeX) Mar 25, 97

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Abstract. We study the deterministic and random Ruelle transfer operator $\cL$ induced by an expanding map $f$ of a smooth $n$-dimensional manifold $X$ and a bundle automorphism $\varphi$ of an $\m$-dimensional vector bundle $E$. We prove the following exact formula for the essential spectral radius of $\cL$ on the space $\cka$ of $\bk$-times continuously differentiable sections of $E$ with $\alpha$-H\"{o}lder $\bk$-th derivative: \[\ress(\cL;\cka)=\exp\left(\sup_{\nu\in\erg} \{h_\nu+\lambda_\nu-(\bk+\alpha)\chi_\nu\}\right).\] Here $\erg$ is the set of $f$-ergodic measures, $h_\nu$ is the entropy of $f$ with respect to $\nu$, $\lambda_\nu$ is the largest Lyapunov-Oseledets exponent of the cocycle $\varphi^k(x)=\varphi (f^{k-1}x)\cdot\ldots\cdot\varphi(x)$, and $\chi_\nu$ is the smallest Lyapunov-Oseledets exponent of the differential $Df^k(x)$, $x\in X$, $k=1,2,\ldots$. A similar result holds for the random case.

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