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A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces. (English) Zbl 1140.37307
Summary: We study Ruelle’s transfer operator \(\mathcal{L}\) induced by a \(C^{r+1}\)-smooth expanding map \(\varphi\) of a smooth manifold and a \(C^r\)-smooth bundle automorphism \(\Phi\) of a real vector bundle \(\mathcal{E}\). We prove the following exact formula for the essential spectral radius of \(\mathcal{L}\) on the space \(C^{r,\alpha}\) of \(r\)-times continuously differentiable sections of \(\mathcal{E}\) with \(\alpha\)-Hölder \(r\)th derivative: \(r_{ess}(\mathcal{L};C^{r,\alpha})=\exp\Big(\sup_{\nu\in\text{Erg}} \{h_\nu+\lambda_\nu-(r+\alpha)\chi_\nu\}\Big)\), where Erg is the set of \(\varphi\)-ergodic measures, \(h_\nu\) the entropy of \(\varphi\) with respect to \(\nu\), \(\lambda_\nu\) the largest Lyapunov exponent of the cocycle induced by \(\Phi\), and \(\chi_\nu\) the smallest Lyapunov exponent for the differential \(D\varphi\).

MSC:
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
47B38 Linear operators on function spaces (general)
37A05 Dynamical aspects of measure-preserving transformations
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