Sharp estimates in Ruelle theorems for matrix transfer operators
In the paper spectral properties of a matrix coefficient transfer operator \(L\) are studied. The operator \(L\) acts on the space of smooth sections of an \(m\)-dimensional real vector bundle over an \(n\)-dimensional compact manifold \(X\) by the formula \((L\phi) (x)= \sum_{y \in f^{-1} (x)} \phi(y) \phi(y)\), \(\phi \in C^r\), \(x\in X\), where \(f: X\to X\) is a smooth covering and \(\phi\) is a smooth bundle automorphism. Let \(M\) be the set of ergodic \(f\)-invariant Borel probability measures on \(X\), \(h_\nu\) be the entropy of \(f\) with respect to \(\nu \in M\), \(\lambda_\nu\) denote the largest Lyapunov-Oseledec exponent of the cocycle \(\phi_k(x) =\phi (f^{k-1}x) \cdots \phi (x)\) over \(f\) and \(\chi_\nu\) denote the smallest Lyapunov-Oseledec exponent of the differential of \(f\). In the paper it is shown that (1) If \(f\) is any smooth covering, than the spectral radius (the essential spectral radius) of \(L\) is less or equal to \(\exp (\sup_{x \in M} \{h_\nu +\lambda_\nu\})\) \((\exp (\sup_{\nu \in M} \{h_\nu +\lambda_\nu -r\chi_\nu\})\). (2) If \(f\) is expanding, than the spectral radius of \(L\) on \(C^0\) is equal to \(\exp (\sup_{\nu \in M} \{h_\nu +\lambda_\nu\})\) and the approximate point spectrum of \(L\) on \(C^0\) contains the annulus \(\{x \in C: \exp (\sup_{\nu\in M} \lambda_\nu) \leq|x|\leq\exp (\sup_{\nu \in M} \{h_\nu+ \lambda_\nu\})\}\). (2) remains correct if \((X,f)\) is a topologically mixing subshift of finite type.
scientific article; zbMATH DE number 1036320