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Constant slope, entropy, and horseshoes for a map on a tame graph

Part of: Topological dynamics Low-dimensional dynamical systems

Published online by Cambridge University Press:  22 April 2019

ADAM BARTOŠ Open the ORCID record for ADAM BARTOŠ [Opens in a new window] ,
JOZEF BOBOK Open the ORCID record for JOZEF BOBOK [Opens in a new window] ,
PAVEL PYRIH Open the ORCID record for PAVEL PYRIH [Opens in a new window] ,
SAMUEL ROTH Open the ORCID record for SAMUEL ROTH [Opens in a new window]  and
BENJAMIN VEJNAR Open the ORCID record for BENJAMIN VEJNAR [Opens in a new window]
ADAM BARTOŠ
Affiliation:
Charles University in Prague, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Czech Republic email bartos@karlin.mff.cuni.cz, pyrih@karlin.mff.cuni.cz, vejnar@karlin.mff.cuni.cz
JOZEF BOBOK
Affiliation:
Czech Technical University in Prague, Faculty of Civil Engineering, Czech Republic email jozef.bobok@cvut.cz
PAVEL PYRIH
Affiliation:
Charles University in Prague, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Czech Republic email bartos@karlin.mff.cuni.cz, pyrih@karlin.mff.cuni.cz, vejnar@karlin.mff.cuni.cz
SAMUEL ROTH
Affiliation:
Silesian University in Opava, Mathematical Institute, Czech Republic email samuel.roth@math.slu.cz
BENJAMIN VEJNAR
Affiliation:
Charles University in Prague, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Czech Republic email bartos@karlin.mff.cuni.cz, pyrih@karlin.mff.cuni.cz, vejnar@karlin.mff.cuni.cz
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Abstract

We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map $f$ of a tame graph $G$ is conjugate to a map $g$ of constant slope. In particular, we show that in the case of a Markov map $f$ that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope $e^{h_{\text{top}}(f)}$, where $h_{\text{top}}(f)$ is the topological entropy of $f$. Moreover, we show that in our class the topological entropy $h_{\text{top}}(f)$ is achievable through horseshoes of the map $f$.

Keywords

Markov maptame graphconstant slopeconjugacyentropy

MSC classification

Primary: 37E25: Maps of trees and graphs
Secondary: 37B40: Topological entropy 37B45: Continua theory in dynamics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 2970 - 2994
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Copyright
© Cambridge University Press, 2019

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