A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces

Abstract

We study Ruelle's transfer operator \mathcal{L} induced by a C^{\mathbf{r}+1}-smooth expanding map \varphi of a smooth manifold and a C^{\mathbf{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^{\mathbf{r},\alpha} of \mathbf{r}-times continuously differentiable sections of \mathcal{E} with \alpha-Hölder \mathbf{r}th derivative:

{\rm r}_{\rm ess}(\mathcal{L};C^{\mathbf{r},\alpha})=\exp\Big(\sup_{\nu\in{\rm Erg}} \{h_\nu+\lambda_\nu-(\mathbf{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.