On notions of determinism in topological dynamics
type: Item
instance of: scholarly article
title: On notions of determinism in topological dynamics
zbMATH Open document ID: 1244.37012
DOI: 10.1017/S0143385710000738
author: Michael Hochman
published in: Ergodic Theory and Dynamical Systems
publication date: 2012-05-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0901.3606
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 6038227
zbMATH Keywords: zero entropy
zbMATH Keywords: topologically predictable
zbMATH Keywords: uniquely ergodic
MaRDI profile type: Publication
arXiv ID: 0901.3606
cites work: PREIMAGE ENTROPY FOR MAPPINGS
cites work: An Introduction to Symbolic Dynamics and Coding
cites work: Extreme Relations for Topological Flows
cites work: Entropy and preimage sets
cites work: Q4065756
cites work: Pre-image entropy
cites work: Topological entropy of m-fold maps
cites work: The topological entropy versus level sets for interval maps
OpenAlex ID: W3103963141
review text: In this interesting paper the author investigates relations between determinism in topological dynamics and entropy. A dynamical system \((X,f)\) (i.e., a continuous map \(f: X\to X\) on a compact metric space) is said to be topologically predictable (TP) if every topological factor of \((X,f)\) is invertible. NEWLINENEWLINENEWLINE NEWLINEThe main results of the paper can be summarized as follows: NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[1.]For every ergodic system \((X,\mu,\mathcal{B},T)\) with zero entropy there are a dynamical system \((Y,S)\) and an invariant measure \(\nu\) on \(Y\) such that \((Y,\nu,S)\) is isomorphic to \((X,\mu,\mathcal{B},T)\) and every point in the system \((Y\times Y, S\times S)\) is recurrent (in particular \((Y,S)\) is TP). NEWLINE\item[2.]Every TP system has zero topological entropy. (This was known before; the author provides a new and more direct proof which works also for \(\mathbb{Z}^d\) actions, with appropriately extended definition of TP). NEWLINE\item[3.]Every measure-preserving system with entropy zero is isomorphic to a shift-invariant Borel measure on a uniquely ergodic subshift \(X\subset \{0,1\}^{\mathbb{Z}}\) with the property that every sequence infinite to the left \(x\in \{0,1\}^{-\mathbb{N}}\) has at most two bi-infinite extensions in \(X\).NEWLINENEWLINE\end{itemize}}
reviewed by: Piotr Oprocha
description: scientific article; zbMATH DE number 6038227