Dynamical fibre bundles. Entropy and dimension
type: Item
instance of: scholarly article
title: Dynamical fibre bundles. Entropy and dimension
zbMATH Open document ID: 0758.58017
DOI: 10.1016/S0294-1449(16)30242-6
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1993-01-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78273
review text: The author studies a \(C^{1,t}\)-map \(\varphi\) on a subset \(X\) of a Banach space \(E\) leaving a Borel probability \(\mu\) on \(X\) invariant. The set \(X\) is assumed to have finite fractal dimension. The main theorem provides inequalities relating local entropy, local \(\alpha\)-entropy and the sum of positive Lyapunov exponents. Under the additional assumptions that \(X\) is compact and that the tangent maps of \(\varphi\) are uniformly asymptotically compact, these inequalities imply the following corollary: If the fractal dimension of \(\mu\) and the Lyapunov dimension of the tangent bundle coincide, then a local version of Pesin's formula holds, i.e. \(h_{\mu}(x,\varphi)=\sum_ id_ i(x)\lambda^ +_ i(x)\). One should observe that even in the finite-dimensional case this result has a new aspect: It is not necessary to know a priori that \(\varphi\) is invertible.
reviewed by: G. Keller
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 93900
zbMATH Keywords: dynamical system
zbMATH Keywords: ergodic theory
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: entropy
zbMATH Keywords: nonlinear analysis
zbMATH Keywords: fractal dimension
author: Ph. Thieullen
MaRDI profile type: Publication
cites work: Q3316247
cites work: The metric entropy of diffeomorphisms. II: Relations between entropy, exponents and dimension
cites work: Q3672767
cites work: A proof of Pesin's formula
cites work: CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY
cites work: Infinite-dimensional dynamical systems in mechanics and physics
cites work: Fibres dynamiques asymptotiquement compacts, exposants de Lyapunov. Entropie. Dimension. (Asymptotically compact dynamical bundles, Lyapunov exponents. Entropy. Dimension)
cites work: Q3979040
cites work: Q3931654
description: scientific article; zbMATH DE number 93900