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Topological and algebraic reducibility for patterns on trees

Published online by Cambridge University Press:  13 August 2013

LLUÍS ALSEDÀ,
DAVID JUHER and
FRANCESC MAÑOSAS
LLUÍS ALSEDÀ
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
DAVID JUHER
Affiliation:
Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain email david.juher@udg.edu
FRANCESC MAÑOSAS
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
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Abstract

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We extend the classical notion of block structure for periodic orbits of interval maps to the setting of tree maps and study the algebraic properties of the Markov matrix of a periodic tree pattern having a block structure. We also prove a formula which relates the topological entropy of a pattern having a block structure with that of the underlying periodic pattern obtained by collapsing each block to a point, and characterize the structure of the zero entropy patterns in terms of block structures. Finally, we prove that an $n$-periodic pattern has zero (positive) entropy if and only if all $n$-periodic patterns obtained by considering the $k\mathrm{th} $ iterate of the map on the invariant set have zero (respectively, positive) entropy, for each $k$ relatively prime to $n$.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 1 , February 2015, pp. 34 - 63
Copyright
© Cambridge University Press, 2013 

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