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May  2018, 38(5): 2541-2554. doi: 10.3934/dcds.2018106

Constant slope models for finitely generated maps

Silesian University in Opava, Na Rybničku 626/1, 746 01 Opava, Czech Republic

Received  July 2017 Published  March 2018

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.

Citation: Samuel Roth. Constant slope models for finitely generated maps. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2541-2554. doi: 10.3934/dcds.2018106
References:
[1]

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.

[2]

J. Bobok, Semiconjugacy to a map of a constant slope, Studia Math., 208 (2012), 213-228.  doi: 10.4064/sm208-3-2.

[3]

J. Bobok and H. Bruin, Constant slope maps and the Vere-Jones classification, Entropy, 18 (2016), Paper No. 234, 27 pp.

[4]

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.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. 3rd Ed. Vol. 1. John Wiley & Sons, Inc., New York, 1950.

[6]

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.

[7]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, 1995.

[8]

B. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided, and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998.

[9]

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.

[10]

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.

[11]

S. Ruette, Chaos on the Interval, University Lecture Series, 67. American Mathematical Society, Providence, RI, 2017.

[12]

D. Vere-Jones, Ergodic properties of nonnegative matrices-I, Pacific J. Math., 22 (1967), 361-386.  doi: 10.2140/pjm.1967.22.361.

show all references

References:
[1]

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.

[2]

J. Bobok, Semiconjugacy to a map of a constant slope, Studia Math., 208 (2012), 213-228.  doi: 10.4064/sm208-3-2.

[3]

J. Bobok and H. Bruin, Constant slope maps and the Vere-Jones classification, Entropy, 18 (2016), Paper No. 234, 27 pp.

[4]

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.

[5]

W. Feller, An Introduction to Probability Theory and Its Applications. 3rd Ed. Vol. 1. John Wiley & Sons, Inc., New York, 1950.

[6]

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.

[7]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, UK, 1995.

[8]

B. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided, and Countable State Markov Shifts, Universitext. Springer-Verlag, Berlin, 1998.

[9]

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.

[10]

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.

[11]

S. Ruette, Chaos on the Interval, University Lecture Series, 67. American Mathematical Society, Providence, RI, 2017.

[12]

D. Vere-Jones, Ergodic properties of nonnegative matrices-I, Pacific J. Math., 22 (1967), 361-386.  doi: 10.2140/pjm.1967.22.361.

Figure 1.  The production of a finitely generated map
Table 1.  Some countably monotone maps from the literature -cf. Section 7.2
Graph
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
Graph
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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