# American Institute of Mathematical Sciences

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

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