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Generic chaos on dendrites

Part of: Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  19 March 2021

ĽUBOMÍR SNOHA ,
VLADIMÍR ŠPITALSKÝ Open the ORCID record for VLADIMÍR ŠPITALSKÝ [Opens in a new window]  and
MICHAL TAKÁCS
ĽUBOMÍR SNOHA
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica 974 01, Slovakia (e-mail: lubomir.snoha@umb.sk, michal.takacs.math@gmail.com)
VLADIMÍR ŠPITALSKÝ*
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica 974 01, Slovakia (e-mail: lubomir.snoha@umb.sk, michal.takacs.math@gmail.com)
MICHAL TAKÁCS
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica 974 01, Slovakia (e-mail: lubomir.snoha@umb.sk, michal.takacs.math@gmail.com)
*
e-mail: vladimir.spitalsky@umb.sk
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Abstract

We characterize dendrites D such that a continuous selfmap of D is generically chaotic (in the sense of Lasota) if and only if it is generically ${\varepsilon }$-chaotic for some ${\varepsilon }>0$. In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with non-empty interiors under iterates of the map. A dendrite D belongs to this class if and only if it is completely regular, with all points of finite order (that is, if and only if D contains neither a copy of the Riemann dendrite nor a copy of the $\omega $-star).

Keywords

completely regular continuumdendritegeneric chaosscrambled pair

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37B45: Continua theory in dynamics 37E99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 6 , June 2022 , pp. 2108 - 2150
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Dedicated to the memory of Sylvie Ruette

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