Maximizing entropy of cycles on trees

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

Maximizing entropy of cycles on trees

Lluís Alsedà ^1, , David Juher ^2, , Deborah M. King ^3, and Francesc Mañosas ^1,

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

Abstract
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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".

Keywords: patterns, topological entropy., Tree maps.

Mathematics Subject Classification: Primary: 37E2.

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.

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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.

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