American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcds.2021131
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Forward triplets and topological entropy on trees

Lluís Alsedà 1, , David Juher 2,, and  Francesc Mañosas 3,

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

Received  March 2021 Revised  July 2021 Early access September 2021

Fund Project: 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

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 $.

Keywords: Tree maps, periodic patterns, topological entropy.
Mathematics Subject Classification: Primary: 37E15, 37E25.
Citation: Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021131
References:
[1]

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.  doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[3]

L. Alsedà, D. Juher and F. Mañosas, Topological and algebraic reducibility for patterns on trees, Ergodic Theory Dynam. Systems, 35 (2015), 34-63.  doi: 10.1017/etds.2013.52.  Google Scholar

[4]

L. Alsedà, D. Juher and F. Mañosas, On the minimum positive entropy for cycles on trees, Tran. Amer. Math. Soc., 369 (2017), 187-221.  doi: 10.1090/tran6677.  Google Scholar

[5]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2$^{nd}$ edition, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/4205.  Google Scholar

[6]

L. Alsedà and X. Ye, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.  doi: 10.1017/S0143385700008348.  Google Scholar

[7]

S. Baldwin, Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math., 67 (1987), 111-127.  doi: 10.1016/0012-365X(87)90021-5.  Google Scholar

[8]

F. Blanchard, E. Glasner, S. Kolyada and A. Maas, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

[9]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 819 (1980), 18–34.  Google Scholar

[10]

A. Blokh, Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.  doi: 10.1016/0040-9383(94)90019-1.  Google Scholar

[11]

M. A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc., 224 (2013), 97 pp. doi: 10.1090/S0065-9266-2012-00671-X.  Google Scholar

[12]

T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[13]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.  doi: 10.1016/0040-9383(93)90014-M.  Google Scholar

[14]

M. Misiurewicz, Minor cycles for interval maps, Fund. Math., 145 (1994), 281-304.  doi: 10.4064/fm-145-3-281-304.  Google Scholar

[15]

M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), 112 pp. doi: 10.1090/memo/0456.  Google Scholar

[16]

A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  doi: 10.1142/S0218127495000934.  Google Scholar

[17]

P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248.  doi: 10.1007/BF01614086.  Google Scholar

show all references

References:
[1]

R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

L. Alsedà, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.  doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[3]

L. Alsedà, D. Juher and F. Mañosas, Topological and algebraic reducibility for patterns on trees, Ergodic Theory Dynam. Systems, 35 (2015), 34-63.  doi: 10.1017/etds.2013.52.  Google Scholar

[4]

L. Alsedà, D. Juher and F. Mañosas, On the minimum positive entropy for cycles on trees, Tran. Amer. Math. Soc., 369 (2017), 187-221.  doi: 10.1090/tran6677.  Google Scholar

[5]

L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2$^{nd}$ edition, Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/4205.  Google Scholar

[6]

L. Alsedà and X. Ye, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.  doi: 10.1017/S0143385700008348.  Google Scholar

[7]

S. Baldwin, Generalizations of a theorem of Sarkovskii on orbits of continuous real-valued functions, Discrete Math., 67 (1987), 111-127.  doi: 10.1016/0012-365X(87)90021-5.  Google Scholar

[8]

F. Blanchard, E. Glasner, S. Kolyada and A. Maas, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.  Google Scholar

[9]

L. Block, J. Guckenheimer, M. Misiurewicz and L. S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems, Lecture Notes in Math., Springer, Berlin, 819 (1980), 18–34.  Google Scholar

[10]

A. Blokh, Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.  doi: 10.1016/0040-9383(94)90019-1.  Google Scholar

[11]

M. A. Blokh, R. Fokkink, J. Mayer, L. Oversteegen and E. Tymchatyn, Fixed point theorems for plane continua with applications, Mem. Amer. Math. Soc., 224 (2013), 97 pp. doi: 10.1090/S0065-9266-2012-00671-X.  Google Scholar

[12]

T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992.  doi: 10.1080/00029890.1975.11994008.  Google Scholar

[13]

J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664.  doi: 10.1016/0040-9383(93)90014-M.  Google Scholar

[14]

M. Misiurewicz, Minor cycles for interval maps, Fund. Math., 145 (1994), 281-304.  doi: 10.4064/fm-145-3-281-304.  Google Scholar

[15]

M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), 112 pp. doi: 10.1090/memo/0456.  Google Scholar

[16]

A. N. Sharkovsky, Co-existence of cycles of a continuous mapping of the line into itself, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 1263-1273.  doi: 10.1142/S0218127495000934.  Google Scholar

[17]

P. Štefan, A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Comm. Math. Phys., 54 (1977), 237-248.  doi: 10.1007/BF01614086.  Google Scholar

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} $
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 6.  Two models of the same pattern
Figure Options

Download full-size image

Download as PowerPoint slide

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

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 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Topological induction
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[2]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[3]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[4]

Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
[5]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[6]

José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415
[7]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[8]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629
[9]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235
[10]

Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
[11]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[12]

Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501
[13]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[14]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[15]

Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1
[16]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[17]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[18]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[19]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[20]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (50)
  • HTML views (81)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences