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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc., 254 (1979), 391-398. doi: 10.2307/1998276.  Google Scholar

[11]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992.  Google Scholar

[12]

W. Geller and J. Tolosa, Maximal entropy odd orbit types, Trans. Amer. Math. Soc., 329 (1992), 161-171. doi: 10.2307/2154082.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. M. King, Maximal entropy of permutations of even order, Ergodic Theory Dynam. Systems, 17 (1997), 1409-1417. doi: 10.1017/S0143385797086367.  Google Scholar

[16]

______, Non-uniqueness of even order permutations with maximal entropy, Ergodic Theory Dynam. Systems, 20 (2000), 801-807. doi: 10.1017/S0143385700000420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), vi+112.  Google Scholar

[20]

W. Rudin, "Principles of Mathematical Analysis," third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

L. Block, Simple periodic orbits of mappings of the interval, Trans. Amer. Math. Soc., 254 (1979), 391-398. doi: 10.2307/1998276.  Google Scholar

[11]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension," Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992.  Google Scholar

[12]

W. Geller and J. Tolosa, Maximal entropy odd orbit types, Trans. Amer. Math. Soc., 329 (1992), 161-171. doi: 10.2307/2154082.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

D. M. King, Maximal entropy of permutations of even order, Ergodic Theory Dynam. Systems, 17 (1997), 1409-1417. doi: 10.1017/S0143385797086367.  Google Scholar

[16]

______, Non-uniqueness of even order permutations with maximal entropy, Ergodic Theory Dynam. Systems, 20 (2000), 801-807. doi: 10.1017/S0143385700000420.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

M. Misiurewicz and Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc., 94 (1991), vi+112.  Google Scholar

[20]

W. Rudin, "Principles of Mathematical Analysis," third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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