Computing the topological entropy for piecewise monotonic maps on the interval
type: Item
instance of: scholarly article
title: Computing the topological entropy for piecewise monotonic maps on the interval
zbMATH Open document ID: 0938.37004
DOI: 10.1023/A:1004585613252
author: Thomas Steinberger
published in: Journal of Statistical Physics
publication date: 1999-11-22T00:00:00Z
review text: The topological entropy of a map is one of the quantitative measures of the complexity of a dynamical system. This paper deals with a new method for computing the topological entropy of a piecewise monotonic transformation on the interval. For a piecewise monotonic map \(T\) on the interval one can define the topological entropy \(h_{\text{top}}(T)\) by \[ h_{\text{top}}(T): =\lim_{n\to \infty}{1\over n}\log \bigl(c_n(T)\bigr). \] The author shows that \(c_n(T)=c_ne^{h_{\text{top}}(T)\cdot n} +r_n\), where the \(c_n\) are bounded and periodic and \(|r_n|\leq K\alpha^n\) for a constant \(K\) and \(\alpha<e^{h_{\text{top}}(T)}\). The proof of this fact is based on a transition matrix associated with \(T\). For this matrix the author gives a spectral theorem. This is used for an estimation of the accuracy of an algorithm.
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37E05
zbMATH DE Number: 1366060
zbMATH Keywords: interval map
zbMATH Keywords: piecewise-monotonic map
zbMATH Keywords: topological entropy
zbMATH Keywords: piecewise monotonic transformation
zbMATH Keywords: transition matrix
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: a:1004585613252
OpenAlex ID: W1615260365
description: scientific article; zbMATH DE number 1366060