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The reversibility problem for quasi-homogeneous dynamical systems
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 |
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. |
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