95-406 Campbell J., Latushkin Y.

Sharp Estimates in Ruelle Theorems for Matrix Transfer Operators (55K, LaTeX) Sep 1, 95

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Abstract. A matrix coefficient transfer operator $(\calL\Phi)(x)=\sum\phi(y)\Phi(y)$, $y\in f^{-1}(x)$ on the space of $C^r$-sections of an $m$-dimensional vector bundle over $n$-dimensional compact manifold is considered. The spectral radius of $\calL$ is estimated by \newline $\displaystyle{\exp \sup \{ h_\nu + \lambda_\nu:\nu\in\calM\}}$ and the essential spectral radius by \[\exp\sup\{h_\nu+\lambda_\nu-r\cdot\chi_\nu:\nu\in\calM\}.\] Here $\calM$ is the set of ergodic $f$-invariant measures, and for $\nu \in {\cal M}, \; h_{\nu}$ is the measure-theoretic entropy of $f$, $\lambda_\nu$ is the largest Lyapunov exponent of the cocycle over $f$ generated by $\phi$, and $\chi_\nu$ is the smallest Lyapunov exponent of the differential of $f$.

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