The dynamics of homeomorphisms of hereditarily decomposable chainable continua
type: Item
instance of: scholarly article
title: The dynamics of homeomorphisms of hereditarily decomposable chainable continua
zbMATH Open document ID: 0831.54019
DOI: 10.1016/0166-8641(94)00091-G
author: Xiangdong Ye
published in: Topology and its Applications
publication date: 1995-09-04T00:00:00Z
review text: A continuum \(M\) is said to be hereditarily decomposable if each subcontinuum can be decomposed as the union of two proper subcontinua and is said to be chainable (respectively circle-like) if for each \(\varepsilon > 0\), there exists a continuous map \(f_\varepsilon : M \to [0,1]\) (respectively \(S^1\)) such that \(\text{diam} (f_\varepsilon^{-1} (t)) < \varepsilon\) for each \(t\). The author shows that if \(M\) is a hereditarily decomposable chainable continuum and \(F : M \to M\) is a homeomorphism, then any periodic orbit of \(F\) has period a power of 2 and for each \(x \in M\) which is recurrent, the \(\omega\)-limit set of \(x\) is either a periodic orbit of \(F\) or is a subset of \(M\) on which \(F\) acts as a dyadic adding machine. The author deduces that homeomorphisms of chainable and circle-like Suslinean continua have zero topological entropy in support of a conjecture of M. Barge that all homeomorphisms of all Suslinean continua have zero topological entropy.
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 792533
zbMATH Keywords: hereditarily decomposable chainable continuum
zbMATH Keywords: periodic orbit
zbMATH Keywords: circle- like Suslinean continua
zbMATH Keywords: topological entropy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 792533