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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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