Semiconjugacy to a map of a constant slope

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December 2015, 20(10): 3403-3413. doi: 10.3934/dcdsb.2015.20.3403

Semiconjugacy to a map of a constant slope

Lluís Alsedà ^1, and Michał Misiurewicz ^2,

1.	Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona
2.	Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202

Received October 2014 Revised March 2015 Published September 2015

Abstract
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It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone graph maps. We show that assigning to a continuous transitive piecewise monotone map of positive entropy a map of constant slope conjugate to it defines an operator, and show that this operator is not continuous.

Keywords: interval Markov maps, topological entropy, semiconjugacy to a map of constant slope, Piecewise monotonotone maps, measure of maximal entropy..

Mathematics Subject Classification: Primary: 37E05, 37B4.

Citation: Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403

References:

[1]	Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second Edition, Advanced Series in Nonlinear Dynamics 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/4205.
[2]	A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk., 37 (1982), 189-190.
[3]	M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math., 97 (1990), 27-36.
[4]	F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, Israel J. Math., 38 (1981), 107-115. doi: 10.1007/BF02761854.
[5]	T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192. doi: 10.1090/S0002-9947-1978-0457679-0.
[6]	J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664. doi: 10.1016/0040-9383(93)90014-M.
[7]	J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 465-563. doi: 10.1007/BFb0082847.
[8]	M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.
[9]	M. Misiurewicz, Possible jumps of entropy for interval maps, Qualit. Th. Dyn. Sys., 2 (2001), 289-306. doi: 10.1007/BF02969344.
[10]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[11]	M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps, in From Phase Transitions to Chaos, World Sci. Publ., River Edge, NJ, 1992, 489-500.
[12]	W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.
[13]	P. Raith, Hausdorff dimension for piecewise monotonic maps, Studia Math., 94 (1989), 17-33.
[14]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.

show all references

References:

[1]	Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second Edition, Advanced Series in Nonlinear Dynamics 5, World Scientific Publishing Co. Inc., River Edge, NJ, 2000. doi: 10.1142/4205.
[2]	A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk., 37 (1982), 189-190.
[3]	M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math., 97 (1990), 27-36.
[4]	F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II, Israel J. Math., 38 (1981), 107-115. doi: 10.1007/BF02761854.
[5]	T.-Y. Li and J. A. Yorke, Ergodic transformations from an interval into itself, Trans. Amer. Math. Soc., 235 (1978), 183-192. doi: 10.1090/S0002-9947-1978-0457679-0.
[6]	J. Llibre and M. Misiurewicz, Horseshoes, entropy and periods for graph maps, Topology, 32 (1993), 649-664. doi: 10.1016/0040-9383(93)90014-M.
[7]	J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 465-563. doi: 10.1007/BFb0082847.
[8]	M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.
[9]	M. Misiurewicz, Possible jumps of entropy for interval maps, Qualit. Th. Dyn. Sys., 2 (2001), 289-306. doi: 10.1007/BF02969344.
[10]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[11]	M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps, in From Phase Transitions to Chaos, World Sci. Publ., River Edge, NJ, 1992, 489-500.
[12]	W. Parry, Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc., 122 (1966), 368-378. doi: 10.1090/S0002-9947-1966-0197683-5.
[13]	P. Raith, Hausdorff dimension for piecewise monotonic maps, Studia Math., 94 (1989), 17-33.
[14]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.

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