Forward triplets and topological entropy on trees

Lluís Alsedà; David Juher; Francesc Mañosas

doi:10.3934/dcds.2021131

Article Contents

2022, Volume 42, Issue 2: 623-641. Doi: 10.3934/dcds.2021131

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Forward triplets and topological entropy on trees

1.
Departament de Matemàtiques and Centre de Recerca Matemàtica, Edifici Ciències, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona 08913, Spain
2.
Departament Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, c/ Universitat de Girona, 6, Girona 17003, Spain
3.
Departament de Matemàtiques, Edifici Ciències, Universitat Autònoma de Barcelona, Cerdanyola del Vallès, Barcelona 08913, Spain

^* Corresponding author: David Juher
^* Corresponding author: David Juher

Received: March 2021

Revised: July 2021

Early access: September 2021

Published: February 2022

Work supported by grants MTM2017-86795-C3-1-P and 2017 SGR 1617. Lluís Alsedà acknowledges financial support from the Spanish Ministerio de Economía y Competitividad grant number MDM-2014-0445 within the "María de Maeztu" excellence program

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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 $ f^k $ has a periodic orbit with three aligned points consecutive in time, that is, a triplet $ (a,b,c) $ such that $ f^k(a) = b $, $ f^k(b) = c $ and $ b $ belongs to the interior of the unique interval connecting $ a $ and $ c $ (a forward triplet of $ f^k $). We also prove a new criterion of entropy zero for simplicial $ n $-periodic patterns $ P $ based on the non existence of forward triplets of $ f^k $ for any $ 1\le k<n $ inside $ P $. Finally, we study the set $ \mathcal{X}_n $ 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 $ \mathcal{X}_n $ and prove that this entropy is the unique real root in $ (1,\infty) $ of the polynomial $ x^n-2x-1 $.

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Mathematics Subject Classification: Primary: 37E15, 37E25.

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Figure 1. Two non-homeomorphic trees $ T $ and $ S $, with 9-periodic orbits $ P = \{x_i\}_{i = 1}^9 $ and $ Q = \{y_i\}_{i = 1}^9 $ of two respective (unspecified) tree maps $ {f}: {T} \longrightarrow {T} $ and $ {g}: {S} \longrightarrow {S} $

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Figure 2. An interval model $ (T,P,f) $ and the corresponding pattern, that can be identified with the permutation $ (3,4,2,5,1) $

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Figure 3. Two 8-periodic simplicial patterns $ \mathcal{P} $ and $ \mathcal{Q} $

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Figure 4. A fully rotational 12-periodic pattern $ \mathcal{P} $

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Figure 5. The pattern $ \mathcal{Q}_6 $

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Figure 6. Two models of the same pattern

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Figure 7. On the left, the canonical model $ (T,P,f) $ of a 6-periodic pattern $ \mathcal{P} $, for which $ f(y) = y $. On the right, the Markov graph of $ (T,P,f) $

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Figure 8. The two possible arrangements of the connected components $ C_0,C_1,C_2 $ in the proof of Lemma 4.2. The black points belong to $ P $

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Figure 9. Topological induction

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Figure 10. A 15-periodic pattern with two discrete components and a $ \pi $-reduced 5-pattern

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Figure 11. A 15-periodic pattern with two discrete components and a 3-division

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Figure 12. Two possible sequences $ \mathcal{P}\ge\mathcal{Q}_1\ge\mathcal{Q}_2 $ and $ \mathcal{P}\ge\mathcal{R}_1\ge\mathcal{R}_2 $ of pull outs leading to two (different) complete openings of $ \mathcal{P} $ with respect to the forward triplet $ (4,5,6) $

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