Sharp estimates in Ruelle theorems for matrix transfer operators
type: Item
instance of: scholarly article
title: Sharp estimates in Ruelle theorems for matrix transfer operators
zbMATH Open document ID: 0883.58029
author: O. Diekmann
published in: Communications in Mathematical Physics
publication date: 1998-03-11T00:00:00Z
review text: 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.
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1036320
zbMATH Keywords: matrix transfer operators
zbMATH Keywords: expanding map
zbMATH Keywords: Lyapounov exponent
zbMATH Keywords: cocycle
zbMATH Keywords: entropy
zbMATH Keywords: spectral radius
MaRDI profile type: Publication
full work available at URL: s002200050095
OpenAlex ID: W2059240934
DOI: 10.1007/S002200050095
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description: scientific article; zbMATH DE number 1036320