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 & Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237
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show all references

References:
[1]

Trans. Amer. Math. Soc., 114 (1965), 309-319.  Google Scholar

[2]

Proc. London Math. Soc. (3), 91 (2005), 414-442. doi: 10.1112/S0024611505015224.  Google Scholar

[3]

Topology, 36 (1997), 1123-1153. doi: 10.1016/S0040-9383(96)00039-0.  Google Scholar

[4]

Experiment. Math., 17 (2008), 391-407.  Google Scholar

[5]

second ed., Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.  Google Scholar

[6]

Ergodic Theory Dynam. Systems, 20 (2000), 1559-1576. doi: 10.1017/S0143385700000857.  Google Scholar

[7]

Acta Math. Univ. Comenian. (N.S.), 62 (1993), 237-248.  Google Scholar

[8]

Ergodic Theory Dynam. Systems, 15 (1995), 221-237. doi: 10.1017/S0143385700008348.  Google Scholar

[9]

Discrete Math., 67 (1987), 111-127. doi: 10.1016/0012-365X(87)90021-5.  Google Scholar

[10]

Trans. Amer. Math. Soc., 254 (1979), 391-398. doi: 10.2307/1998276.  Google Scholar

[11]

Lecture Notes in Mathematics, 1513, Springer-Verlag, Berlin, 1992.  Google Scholar

[12]

Trans. Amer. Math. Soc., 329 (1992), 161-171. doi: 10.2307/2154082.  Google Scholar

[13]

Proc. Amer. Math. Soc., 123 (1995), 1917-1922. doi: 10.2307/2161011.  Google Scholar

[14]

Proc. Amer. Math. Soc., 126 (1998), 3709-3713. doi: 10.1090/S0002-9939-98-04493-1.  Google Scholar

[15]

Ergodic Theory Dynam. Systems, 17 (1997), 1409-1417. doi: 10.1017/S0143385797086367.  Google Scholar

[16]

Ergodic Theory Dynam. Systems, 20 (2000), 801-807. doi: 10.1017/S0143385700000420.  Google Scholar

[17]

Qual. Theory Dyn. Syst., 4 (2003), 77-97. doi: 10.1007/BF02972824.  Google Scholar

[18]

Trans. Amer. Math. Soc., 273 (1982), 191-199. doi: 10.2307/1999200.  Google Scholar

[19]

Mem. Amer. Math. Soc., 94 (1991), vi+112.  Google Scholar

[20]

third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics.  Google Scholar

[21]

Ukrain. Mat. Ž., 16 (1964), 61-71.  Google Scholar

[22]

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]

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]

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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