# American Institute of Mathematical Sciences

August  2013, 33(8): 3237-3276. doi: 10.3934/dcds.2013.33.3237

## Maximizing entropy of cycles on trees

 1 Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain, Spain 2 Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain 3 Departament of Mathematics and Statistics, The University of Melbourne, Vic, 3010, Australia

Received  May 2012 Revised  September 2012 Published  January 2013

In this paper we give a partial characterization of the periodic tree patterns of maximum entropy for a given period. More precisely, we prove that each periodic pattern with maximal entropy is irreducible (has no block structures) and simplicial (any vertex belongs to the periodic orbit). Moreover, we also prove that it is maximodal in the sense that every point of the periodic orbit is a "turning point".
Citation: Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237
##### References:
 [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. [2] Ll. Alsedà, F. Gautero, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Patterns and minimal dynamics for graph maps, Proc. London Math. Soc. (3), 91 (2005), 414-442. doi: 10.1112/S0024611505015224. [3] Ll. 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. [4] Ll. Alsedà, D. Juher and D. M. King, A lower bound for the maximum topological entropy of $(4k+2)$-cycles, Experiment. Math., 17 (2008), 391-407. [5] Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," second ed., Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. [6] Ll. Alsedà, F. Mañosas and P. Mumbrú, Minimizing topological entropy for continuous maps on graphs, Ergodic Theory Dynam. Systems, 20 (2000), 1559-1576. doi: 10.1017/S0143385700000857. [7] Ll. Alsedà and X. Ye, Division for star maps with the branching point fixed, Acta Math. Univ. Comenian. (N.S.), 62 (1993), 237-248. [8] ______, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237. doi: 10.1017/S0143385700008348. [9] 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. [10] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc., 254 (1979), 391-398. doi: 10.2307/1998276. [11] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. [12] W. Geller and J. Tolosa, Maximal entropy odd orbit types, Trans. Amer. Math. Soc., 329 (1992), 161-171. doi: 10.2307/2154082. [13] W. Geller and B. Weiss, Uniqueness of maximal entropy odd orbit types, Proc. Amer. Math. Soc., 123 (1995), 1917-1922. doi: 10.2307/2161011. [14] W. Geller and Z. Zhang, Maximal entropy permutations of even size, Proc. Amer. Math. Soc., 126 (1998), 3709-3713. doi: 10.1090/S0002-9939-98-04493-1. [15] D. M. King, Maximal entropy of permutations of even order, Ergodic Theory Dynam. Systems, 17 (1997), 1409-1417. doi: 10.1017/S0143385797086367. [16] ______, Non-uniqueness of even order permutations with maximal entropy, Ergodic Theory Dynam. Systems, 20 (2000), 801-807. doi: 10.1017/S0143385700000420. [17] D. M. King and J. B. Strantzen, Classification of permutations and cycles of maximum topological entropy, Qual. Theory Dyn. Syst., 4 (2003), 77-97. doi: 10.1007/BF02972824. [18] T. Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc., 273 (1982), 191-199. doi: 10.2307/1999200. [19] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), vi+112. [20] W. Rudin, "Principles of Mathematical Analysis," third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics. [21] O. M. Šarkovs$'$kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., 16 (1964), 61-71. [22] D. Serre, "Matrices," second ed., Graduate Texts in Mathematics, 216, Springer, New York, 2010, Theory and applications. doi: 10.1007/978-1-4419-7683-3. [23] A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Thirty Years After Sharkovskiĭ's Theorem: New Perspectives (Murcia, 1994), World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 8, World Sci. Publ., River Edge, NJ, 1995, Translated by J. Tolosa, Reprint of the paper reviewed in MR1361914 (96j:58058), 1-11. [24] R. S. Varga, "Matrix Iterative Analysis," expanded ed., Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.

show all references

##### References:
 [1] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. [2] Ll. Alsedà, F. Gautero, J. Guaschi, J. Los, F. Mañosas and P. Mumbrú, Patterns and minimal dynamics for graph maps, Proc. London Math. Soc. (3), 91 (2005), 414-442. doi: 10.1112/S0024611505015224. [3] Ll. 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. [4] Ll. Alsedà, D. Juher and D. M. King, A lower bound for the maximum topological entropy of $(4k+2)$-cycles, Experiment. Math., 17 (2008), 391-407. [5] Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," second ed., Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. [6] Ll. Alsedà, F. Mañosas and P. Mumbrú, Minimizing topological entropy for continuous maps on graphs, Ergodic Theory Dynam. Systems, 20 (2000), 1559-1576. doi: 10.1017/S0143385700000857. [7] Ll. Alsedà and X. Ye, Division for star maps with the branching point fixed, Acta Math. Univ. Comenian. (N.S.), 62 (1993), 237-248. [8] ______, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237. doi: 10.1017/S0143385700008348. [9] 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. [10] L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc., 254 (1979), 391-398. doi: 10.2307/1998276. [11] L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992. [12] W. Geller and J. Tolosa, Maximal entropy odd orbit types, Trans. Amer. Math. Soc., 329 (1992), 161-171. doi: 10.2307/2154082. [13] W. Geller and B. Weiss, Uniqueness of maximal entropy odd orbit types, Proc. Amer. Math. Soc., 123 (1995), 1917-1922. doi: 10.2307/2161011. [14] W. Geller and Z. Zhang, Maximal entropy permutations of even size, Proc. Amer. Math. Soc., 126 (1998), 3709-3713. doi: 10.1090/S0002-9939-98-04493-1. [15] D. M. King, Maximal entropy of permutations of even order, Ergodic Theory Dynam. Systems, 17 (1997), 1409-1417. doi: 10.1017/S0143385797086367. [16] ______, Non-uniqueness of even order permutations with maximal entropy, Ergodic Theory Dynam. Systems, 20 (2000), 801-807. doi: 10.1017/S0143385700000420. [17] D. M. King and J. B. Strantzen, Classification of permutations and cycles of maximum topological entropy, Qual. Theory Dyn. Syst., 4 (2003), 77-97. doi: 10.1007/BF02972824. [18] T. Y. Li, M. Misiurewicz, G. Pianigiani and J. A. Yorke, No division implies chaos, Trans. Amer. Math. Soc., 273 (1982), 191-199. doi: 10.2307/1999200. [19] M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), vi+112. [20] W. Rudin, "Principles of Mathematical Analysis," third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics. [21] O. M. Šarkovs$'$kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. Ž., 16 (1964), 61-71. [22] D. Serre, "Matrices," second ed., Graduate Texts in Mathematics, 216, Springer, New York, 2010, Theory and applications. doi: 10.1007/978-1-4419-7683-3. [23] A. N. Sharkovskiĭ, Coexistence of cycles of a continuous map of the line into itself, Thirty Years After Sharkovskiĭ's Theorem: New Perspectives (Murcia, 1994), World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 8, World Sci. Publ., River Edge, NJ, 1995, Translated by J. Tolosa, Reprint of the paper reviewed in MR1361914 (96j:58058), 1-11. [24] R. S. Varga, "Matrix Iterative Analysis," expanded ed., Springer Series in Computational Mathematics, 27, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2.
 [1] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete and 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 and Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . 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 and 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 and 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 and 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 and Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415 [7] Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235 [8] Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127 [9] 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 [10] Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete and Continuous Dynamical Systems, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501 [11] Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 [12] Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 [13] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [14] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [15] 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 [16] 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 [17] Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 [18] Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086 [19] Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 [20] Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041

2020 Impact Factor: 1.392