FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
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...
| Authors: | , , |
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| Format: | article |
| Status: | Versión aceptada para publicación |
| Publication Date: | 2022 |
| Country: | España |
| Institution: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repository: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:2072/535421 |
| Online Access: | http://hdl.handle.net/2072/535421 |
| Access Level: | Open access |
| Keyword: | Periodic patterns Topological entropy Tree maps |
| Summary: | 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. © 2022 American Institute of Mathematical Sciences. All rights reserved. |
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