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Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites

Published online by Cambridge University Press:  02 February 2010

ZDENĚK KOČAN ,
VERONIKA KORNECKÁ-KURKOVÁ  and
MICHAL MÁLEK
ZDENĚK KOČAN
Affiliation:
Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 OPAVA, Czech Republic (email: [email protected], [email protected], [email protected])
VERONIKA KORNECKÁ-KURKOVÁ
Affiliation:
Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 OPAVA, Czech Republic (email: [email protected], [email protected], [email protected])
MICHAL MÁLEK
Affiliation:
Mathematical Institute, Silesian University in Opava, Na Rybníčku 1, 746 01 OPAVA, Czech Republic (email: [email protected], [email protected], [email protected]) Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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Abstract

It is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 165 - 175
Copyright
Copyright © Cambridge University Press 2009

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