Non-existence of a universal zero-entropy system
type: Item
instance of: scholarly article
title: Non-existence of a universal zero-entropy system
zbMATH Open document ID: 1347.37034
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: The author studies the problem of the existence of universal dynamical systems with zero entropy. A topological dynamical system \((X,T)\) is universal in a class \(\mathcal S\) of measure-theoretic dynamical systems when \((X,\mu,T)\) belongs to \(\mathcal S\) for any \(T\)-invariant measure \(\mu\) and if for a given measure-theoretic system in \(\mathcal S\) there exists a measure \(T\)-invariant \(\mu\) such that \((X,\mu,T)\) is measure-theoretic isomorphic to that given system. The main result of this reviewed article states that there does not exist a topological dynamical system which is universal in the class of automorphisms with zero entropy whose topological entropy is equal to zero. Important ingredients for the proof are the notions of measure-theoretic complexity and the symbolic complexity of the system, and when the complexity is dominated by a sequence of numbers. One of the key arguments for the proof (Lemma 2) is about the existence of dynamical systems with zero entropy with the complexity dominated for a given non-decreasing sequence of numbers.
reviewed by: Alejandro Mesón
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37A35
zbMATH DE Number: 6176549
author: J. Serafin
MaRDI profile type: Publication
full work available at URL: s11856-011-0219-x
OpenAlex ID: W2094742397
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DOI: 10.1007/S11856-011-0219-X
description: scientific article; zbMATH DE number 6176549