 97149 Gundlach V. M., Latushkin Y.
 A formula for the essential spectral radius of Ruelle's transfer operator on
smooth H\"older spaces
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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
LyapunovOseledets exponent of the cocycle $\varphi^k(x)=\varphi
(f^{k1}x)\cdot\ldots\cdot\varphi(x)$, and $\chi_\nu$ is the smallest
LyapunovOseledets 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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