# American Institute of Mathematical Sciences

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. [2] A. M. Blokh, Sensitive mappings of an interval,, Uspekhi Mat. Nauk., 37 (1982), 189. [3] M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings,, Studia Math., 97 (1990), 27. [4] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II,, Israel J. Math., 38 (1981), 107. 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. 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. 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, (1342), 1986. 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. [9] M. Misiurewicz, Possible jumps of entropy for interval maps,, Qualit. Th. Dyn. Sys., 2 (2001), 289. doi: 10.1007/BF02969344. [10] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. [11] M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps,, in From Phase Transitions to Chaos, (1992), 489. [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. [13] P. Raith, Hausdorff dimension for piecewise monotonic maps,, Studia Math., 94 (1989), 17. [14] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).

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. [2] A. M. Blokh, Sensitive mappings of an interval,, Uspekhi Mat. Nauk., 37 (1982), 189. [3] M. Denker, G. Keller and M. Urbañski, On the uniqueness of equilibrium states for piecewise monotone mappings,, Studia Math., 97 (1990), 27. [4] F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy II,, Israel J. Math., 38 (1981), 107. 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. 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. 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, (1342), 1986. 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. [9] M. Misiurewicz, Possible jumps of entropy for interval maps,, Qualit. Th. Dyn. Sys., 2 (2001), 289. doi: 10.1007/BF02969344. [10] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings,, Studia Math., 67 (1980), 45. [11] M. Misiurewicz and K. Ziemian, Horseshoes and entropy for piecewise continuous piecewise monotone maps,, in From Phase Transitions to Chaos, (1992), 489. [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. [13] P. Raith, Hausdorff dimension for piecewise monotonic maps,, Studia Math., 94 (1989), 17. [14] P. Walters, An Introduction to Ergodic Theory,, Graduate Texts in Mathematics, (1982).
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