Entropy bounds for endomorphisms commuting with \(k\) actions
type: Item
instance of: scholarly article
title: Entropy bounds for endomorphisms commuting with \(k\) actions
zbMATH Open document ID: 0931.22004
DOI: 10.1007/BF02773458
published in: Israel Journal of Mathematics
publication date: 2000-02-23T00:00:00Z
review text: Let \(\Sigma=\{0,1,\dots,k-1\}^{\mathbb Z^2}\) be the full two-dimensional shift on \(k\) symbols, and let \(h: \Sigma\to\Sigma\) be a shift commuting homeomorphism on \(\Sigma\). \textit{M. Shereshevsky} [Indag. Math., New Ser. 4, 203--210 (1993; Zbl 0794.28010)] showed that \(h\) cannot be expansive and conjectured that \(h\) cannot have finite positive entropy. The authors formulate an algebraic analogue of this problem and prove the following special case of it. Let \(X\) be a compact metrizable abelian group. Let \(T: X\to X\) be a mixing endomorphism. Assume that \(T\) commutes with a \(\mathbb Z^2\)-action with completely positive entropy. Then \(T\) has infinite entropy.
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1216483
zbMATH Keywords: shift commuting homeomorphism
zbMATH Keywords: entropy
zbMATH Keywords: \(\mathbb Z^2\)-action
author: Gary J. Morris
author: Thomas B. Ward
MaRDI profile type: Publication
Wikidata QID: Q61835176
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description: scientific article; zbMATH DE number 1216483