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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

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 & 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.  Google Scholar

[2]

A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk., 37 (1982), 189-190.  Google Scholar

[3]

M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math., 97 (1990), 27-36.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[8]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.  Google Scholar

[9]

M. Misiurewicz, Possible jumps of entropy for interval maps, Qualit. Th. Dyn. Sys., 2 (2001), 289-306. doi: 10.1007/BF02969344.  Google Scholar

[10]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. Raith, Hausdorff dimension for piecewise monotonic maps, Studia Math., 94 (1989), 17-33.  Google Scholar

[14]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.  Google Scholar

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.  Google Scholar

[2]

A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk., 37 (1982), 189-190.  Google Scholar

[3]

M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings, Studia Math., 97 (1990), 27-36.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.  Google Scholar

[9]

M. Misiurewicz, Possible jumps of entropy for interval maps, Qualit. Th. Dyn. Sys., 2 (2001), 289-306. doi: 10.1007/BF02969344.  Google Scholar

[10]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. Raith, Hausdorff dimension for piecewise monotonic maps, Studia Math., 94 (1989), 17-33.  Google Scholar

[14]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York, 1982.  Google Scholar

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