Home > Articles > Published articles > Forward triplets and topological entropy on trees |
Date: | 2022 |
Abstract: | We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map f has positive entropy if and only if some iterate fk has a periodic orbit with three aligned points consecutive in time, that is, a triplet (a,b,c) such that fk(a)=b, fk(b)=c and b belongs to the interior of the unique interval connecting a and c (a forward triplet of fk). We also prove a new criterion of entropy zero for simplicial n-periodic patterns P based on the non existence of forward triplets of fk for any 1≤k<n inside P. Finally, we study the set Xn of all n-periodic patterns P that have a forward triplet inside P. For any n, we define a pattern that attains the minimum entropy in Xn and prove that this entropy is the unique real root in (1,∞) of the polynomial xn−2x−1. |
Grants: | Agencia Estatal de Investigación MTM2017-86795-C3-1-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 Ministerio de Economía y Competitividad MDM-2014-0445 |
Rights: | Tots els drets reservats. |
Language: | Anglès |
Document: | Article ; recerca ; Versió acceptada per publicar |
Subject: | Tree maps ; Periodic patterns ; Topological entropy |
Published in: | Discrete and continuous dynamical systems. Series A, Vol. 42, Issue 2 (February 2022) , p. 623-641, ISSN 1553-5231 |
Postprint 18 p, 409.4 KB |