A splitting theorem for transitive maps
type: Item
instance of: scholarly article
title: A splitting theorem for transitive maps
zbMATH Open document ID: 0959.37032
author: M. A. del Río
author: Lluís Alsedà
author: J. A. Rodríguez
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-04-24T00:00:00Z
review text: The basic result is the following theorem. For a continuous transitive map \(f : X \rightarrow X\), \(X\) a locally connected compact metric space, exactly one of the following two statements holds: (1) \(f\) is totally transitive i.e. \(f^k\) is transitive for all \(k \geq 1\). (2) There exist closed connected subsets \(X_0, \dots X_{k-1}\) of \(X\), \(k > 1\), with nonempty interiors such that \(X = \bigcup_{i=0}^{k-1}X_i\), \(\text{int}(X_i) \cap X_j = \emptyset \), for \(i \neq j\), and \(f(X_i) = X_{i+1\pmod k}\). Moreover, \(f^k|_{X_i}\) is transitive, for \(i = 0, \dots, k-1\). As a consequence, the authors establish lower bounds for the topological entropy of transitive graph maps in terms of the cardinality of their splittings. They also show that for a connected graph which is not a tree, the infimum of the topological entropy of the transitive maps having a periodic point is zero.
Mathematics Subject Classification ID: 37E25
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1286154
zbMATH Keywords: splitting
zbMATH Keywords: transitivity
zbMATH Keywords: topological entropy
reviewed by: Jerzy Ombach
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6277
OpenAlex ID: W2093706895
cites work: Q4858735
cites work: Entropy of transitive tree maps
cites work: Entropy and periodic points for transitive maps
cites work: Chaos, Periodicity, and Snakelike Continua
cites work: Dynamics in one dimension
cites work: On sensitive mappings of the interval
cites work: On the connection between entropy and transitivity for one-dimensional mappings
cites work: Q5422499
cites work: The second iterate of a map with dense orbit
cites work: Exotic attractors. From Liapunov stability to riddled basins
cites work: Ergodic theory on compact spaces
cites work: Q4278138
cites work: Une Minoration De L'Entropie Topologique Des Difféomorphismes Du Disque
cites work: Q3999514
DOI: 10.1006/JMAA.1999.6277
description: scientific article; zbMATH DE number 1286154