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Constant slope models for finitely generated maps

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  • We study countably monotone and Markov interval maps. We establish sufficient conditions for uniqueness of a conjugate map of constant slope. We explain how global window perturbation can be used to generate a large class of maps satisfying these conditions.

    Mathematics Subject Classification: Primary: 37E05; Secondary: 37B40.


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  • Figure 1.  The production of a finitely generated map

    Table 1.  Some countably monotone maps from the literature -cf. Section 7.2

    Source [3,§ 7.1] [4,§ 4] [9,§ 8] [3,§ 7.2.1] [3,§ 7.2.1]
    Vere-Jones classification Recurrent Transient Recurrent Recurrent Transient
    Finitely generated No No No Yes Yes
    Constant slope models None For all $\lambda\geq\exp h(f)$ For all $\lambda\geq\exp h(f)$ Unique, $\lambda=\exp h(f)$ None
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  •   Ll. Alsedà  and  M. Misiurewicz , Semiconjugacy to a map of a constant slope, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015) , 3403-3413.  doi: 10.3934/dcdsb.2015.20.3403.
      J. Bobok , Semiconjugacy to a map of a constant slope, Studia Math., 208 (2012) , 213-228.  doi: 10.4064/sm208-3-2.
      J. Bobok and H. Bruin, Constant slope maps and the Vere-Jones classification, Entropy, 18 (2016), Paper No. 234, 27 pp.
      J. Bobok  and  M. Soukenka , On piecewise affine interval maps with countably many laps, Discrete Cont. Dyn. Syst., 31 (2011) , 753-762.  doi: 10.3934/dcds.2011.31.753.
      W. Feller, An Introduction to Probability Theory and Its Applications. 3rd Ed. Vol. 1. John Wiley & Sons, Inc., New York, 1950.
      T. E. Harris , Transient Markov chains with stationary measures, Proc. Amer. Math. Soc., 8 (1957) , 937-942.  doi: 10.1090/S0002-9939-1957-0091564-3.
      A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, 1995.
      B. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided, and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998.
      M. Misiurewicz  and  S. Roth , No semiconjugacy to a map of constant slope, Ergodic Theory Dynam. Systems, 36 (2016) , 875-889.  doi: 10.1017/etds.2014.81.
      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.
      S. Ruette, Chaos on the Interval, University Lecture Series, 67. American Mathematical Society, Providence, RI, 2017.
      D. Vere-Jones , Ergodic properties of nonnegative matrices-I, Pacific J. Math., 22 (1967) , 361-386.  doi: 10.2140/pjm.1967.22.361.
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