American Institute of Mathematical Sciences

Advanced
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 - A, 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.   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.  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.  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.   Google Scholar

[5]

Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,", second ed., 5 (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.  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.   Google Scholar

[8]

______, No division and the set of periods for tree maps,, Ergodic Theory Dynam. Systems, 15 (1995), 221.  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.  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.  doi: 10.2307/1998276.  Google Scholar

[11]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics, 1513 (1992).   Google Scholar

[12]

W. Geller and J. Tolosa, Maximal entropy odd orbit types,, Trans. Amer. Math. Soc., 329 (1992), 161.  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.  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.  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.  doi: 10.1017/S0143385797086367.  Google Scholar

[16]

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

[20]

W. Rudin, "Principles of Mathematical Analysis,", third ed., (1976).   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.   Google Scholar

[22]

D. Serre, "Matrices,", second ed., 216 (2010).  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, 8 (1994), 1.   Google Scholar

[24]

R. S. Varga, "Matrix Iterative Analysis,", expanded ed., 27 (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.   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.  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.  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.   Google Scholar

[5]

Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One,", second ed., 5 (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.  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.   Google Scholar

[8]

______, No division and the set of periods for tree maps,, Ergodic Theory Dynam. Systems, 15 (1995), 221.  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.  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.  doi: 10.2307/1998276.  Google Scholar

[11]

L. S. Block and W. A. Coppel, "Dynamics in One Dimension,", Lecture Notes in Mathematics, 1513 (1992).   Google Scholar

[12]

W. Geller and J. Tolosa, Maximal entropy odd orbit types,, Trans. Amer. Math. Soc., 329 (1992), 161.  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.  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.  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.  doi: 10.1017/S0143385797086367.  Google Scholar

[16]

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

[20]

W. Rudin, "Principles of Mathematical Analysis,", third ed., (1976).   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.   Google Scholar

[22]

D. Serre, "Matrices,", second ed., 216 (2010).  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, 8 (1994), 1.   Google Scholar

[24]

R. S. Varga, "Matrix Iterative Analysis,", expanded ed., 27 (2000).  doi: 10.1007/978-3-642-05156-2.  Google Scholar

[1]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501
[11]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[12]

Jaume Llibre. Brief survey on the topological entropy. Discrete & 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 & 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 & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[15]

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
[16]

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
[17]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[18]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
[19]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[20]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences