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

Semiconjugacy to a map of a constant slope

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.
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, (2000).  doi: 10.1142/4205.  Google Scholar

[2]

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

[4]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II,, Israel J. Math., 38 (1981), 107.  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.  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.  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, (1342), 1986.  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.   Google Scholar

[9]

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

[10]

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

[11]

M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps,, in From Phase Transitions to Chaos, (1992), 489.   Google Scholar

[12]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368.  doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar

[13]

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

[14]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

show all references

References:
[1]

Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second Edition, (2000).  doi: 10.1142/4205.  Google Scholar

[2]

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

[4]

F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II,, Israel J. Math., 38 (1981), 107.  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.  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.  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, (1342), 1986.  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.   Google Scholar

[9]

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

[10]

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

[11]

M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps,, in From Phase Transitions to Chaos, (1992), 489.   Google Scholar

[12]

W. Parry, Symbolic dynamics and transformations of the unit interval,, Trans. Amer. Math. Soc., 122 (1966), 368.  doi: 10.1090/S0002-9947-1966-0197683-5.  Google Scholar

[13]

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

[14]

P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).   Google Scholar

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