Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents
type: Item
instance of: scholarly article
title: Invariant measures satisfying an equality relating entropy, folding entropy and negative Lyapunov exponents
zbMATH Open document ID: 1173.37018
author: Peidong Liu
published in: Communications in Mathematical Physics
publication date: 2009-03-20T00:00:00Z
review text: The article deals with a connected compact Riemannian manifold \(M\), a \(C^2\) non-invertible map \(f: \;M \to M\) with no critical points, and an \(f\)-invariant probability measure \(\mu\). The author investigates sufficient and necessary conditions for the validity of equality \[ h_\mu(f) = F_\mu(f) - \int \sum_i \lambda_i^-(x) \, d\mu(x),\tag{1} \] where \(h_\mu(f)\) is the measure-theoretic entropy, \(\lambda_i^-(x)\) are all negative Lyapunov exponents at \(x\) (taking into account their multiplicity), and \(F_\mu(f)\) is the folding entropy, which can be defined as \[ F_\mu(f) =-\int \sum_{x\in f^{-1}(y)} p(x)\log p(x)\, d\mu(y), \] where \(p(x)\) is the conditional distribution of \(\mu\) on \(f^{-1}(y)\). For each \(x\), it is considered the stable manifold \(W^s(x)\), i.e., the set of all \(y\) whose orbits \(\{f^n(y)\}\) approach exponentially \(\{f^n(x)\}\). Locally one can define the conditional distributions of \(\mu\) on the stable manifolds. The main result is the following: equality (1) [Commun. Math. Phys. 240, No. 3, 531--538 (2003; Zbl 1078.37024)] holds true if and, under some additional condition, only if \(\mu\) has absolutely continuous conditional distributions on the stable manifolds. The additional condition mentioned above is the Hölder continuity of the Jacobian \(J_f\) computed with respect to \(\mu\). This paper continues previous author's investigations where he proved in a more general setting that \(h_\mu(f)\) does not exceed the right-hand side of (1) see [loc. cit.].
Mathematics Subject Classification ID: 37C40
zbMATH DE Number: 5530491
zbMATH Keywords: invariant measures
zbMATH Keywords: entropy and folding entropy
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: absolutely continuous measure
zbMATH Keywords: Riemannian manifolds
zbMATH Keywords: measure-preserving maps
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
full work available at URL: s00220-008-0568-4
OpenAlex ID: W2060658990
cites work: A proof of the estimation from below in Pesin's entropy formula
cites work: The metric entropy of diffeomorphisms. I: Characterization of measures satisfying Pesin's entropy formula
cites work: Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents
cites work: Pesin’s entropy formula for endomorphisms
cites work: Smooth ergodic theory of random dynamical systems
cites work: A proof of Pesin's formula
cites work: Q5565255
cites work: SRB measures and Pesin's entropy formula for endomorphisms
cites work: LECTURES ON THE ENTROPY THEORY OF MEASURE-PRESERVING TRANSFORMATIONS
cites work: Positivity of entropy production in nonequilibrium statistical mechanics
cites work: An inequality for the entropy of differentiable maps
DOI: 10.1007/S00220-008-0568-4
Recommended article: Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents
Recommended article: The metric entropy of endomorphisms
Recommended article: Lower bound in Pesin formula of C 1 interval maps*
Recommended article: Pesin's entropy formula for systems between \(C^1\) and \(C^{1+\alpha}\)
Recommended article: Ruelle's inequality for the entropy of random diffeomorphisms
Recommended article: Ruelle inequality of folding type for \(C^{1+\alpha}\) maps
Recommended article: Positivity of entropy production in nonequilibrium statistical mechanics
Recommended article: Q5748629
Recommended article: Absolute continuity of hyperbolic invariant measures for endomorphisms
description: scientific article; zbMATH DE number 5530491