February  2022, 42(2): 623-641. doi: 10.3934/dcds.2021131

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

Received  March 2021 Revised  July 2021 Published  February 2022 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 $.

Citation: Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 623-641. doi: 10.3934/dcds.2021131
References:
[1]

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

[2]

L. AlsedàJ. GuaschiJ. LosF. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.  doi: 10.1016/S0040-9383(96)00039-0.

[3]

L. AlsedàD. 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.

[4]

L. AlsedàD. 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.

[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.

[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.

[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.

[8]

F. BlanchardE. GlasnerS. Kolyada and A. Maas, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

[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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

show all references

References:
[1]

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

[2]

L. AlsedàJ. GuaschiJ. LosF. Mañosas and P. Mumbrú, Canonical representatives for patterns of tree maps, Topology, 36 (1997), 1123-1153.  doi: 10.1016/S0040-9383(96)00039-0.

[3]

L. AlsedàD. 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.

[4]

L. AlsedàD. 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.

[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.

[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.

[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.

[8]

F. BlanchardE. GlasnerS. Kolyada and A. Maas, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

[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.

[10]

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

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

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 2.  An interval model $ (T,P,f) $ and the corresponding pattern, that can be identified with the permutation $ (3,4,2,5,1) $
Figure 3.  Two 8-periodic simplicial patterns $ \mathcal{P} $ and $ \mathcal{Q} $
Figure 4.  A fully rotational 12-periodic pattern $ \mathcal{P} $
Figure 5.  The pattern $ \mathcal{Q}_6 $
Figure 6.  Two models of the same pattern
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 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 9.  Topological induction
Figure 10.  A 15-periodic pattern with two discrete components and a $ \pi $-reduced 5-pattern
Figure 11.  A 15-periodic pattern with two discrete components and a 3-division
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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