May  2021, 41(5): 2071-2094. doi: 10.3934/dcds.2020353

Entropy conjugacy for Markov multi-maps of the interval

1. 

Department of Mathematics, Christopher Newport University, Newport News, VA 23606, USA

2. 

Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

* Corresponding author: Kevin McGoff

Received  October 2019 Revised  June 2020 Published  May 2021 Early access  October 2020

We consider a class $ \mathcal{F} $ of Markov multi-maps on the unit interval. Any multi-map gives rise to a space of trajectories, which is a closed, shift-invariant subset of $ [0, 1]^{\mathbb{Z}_+} $. For a multi-map in $ \mathcal{F} $, we show that the space of trajectories is (Borel) entropy conjugate to an associated shift of finite type. Additionally, we characterize the set of numbers that can be obtained as the topological entropy of a multi-map in $ \mathcal{F} $.

Citation: James P. Kelly, Kevin McGoff. Entropy conjugacy for Markov multi-maps of the interval. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2071-2094. doi: 10.3934/dcds.2020353
References:
[1]

E. Akin, The General Topology of Dynamical Systems, volume 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

L. Alvin and J. P. Kelly, Topological entropy of Markov set-valued functions, to appear in Ergodic Theory and Dynamical Systems.  Google Scholar

[3]

L. Alvin and J. P. Kelly, Markov set-valued functions and their inverse limits, Topology Appl., 241 (2018), 102-114.  doi: 10.1016/j.topol.2018.03.035.  Google Scholar

[4]

W. BahsounC. Bose and A. Quas, Deterministic representation for position-dependent random maps, Discrete & Continuous Dynamical Systems - A, 22 (2008), 529-540.  doi: 10.3934/dcds.2008.22.529.  Google Scholar

[5]

I. Banič and T. Lunder, Inverse limits with generalized Markov interval functions, Bull. Malays. Math. Sci. Soc., 39 (2016), 839-848.  doi: 10.1007/s40840-015-0187-0.  Google Scholar

[6]

I. Banič and M. črepnjak, Markov pairs, quasi Markov functions and inverse limits, Houston J. Math., 44 (2018), 695-707.   Google Scholar

[7]

R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar

[8]

R. Bowen, Topological entropy for noncompact sets, Transactions of the American Mathematical Society, 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[9]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

[10]

J. Buzzi, Exponential decay of correlations for random lasota–yorke maps, Communications in mathematical physics, 208 (1999), 25-54.  doi: 10.1007/s002200050746.  Google Scholar

[11]

W. Cordeiro and M. J. Pacífico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar

[12]

M. črepnjak and T. Lunder, Inverse limits with countably Markov interval functions, Glas. Mat. Ser. III, 51 (2016), 491-501.  doi: 10.3336/gm.51.2.14.  Google Scholar

[13]

G. Erceg and J. Kennedy, Topological entropy on closed sets in $[0, 1]^2$, Topology Appl., 246 (2018), 106-136.  doi: 10.1016/j.topol.2018.06.015.  Google Scholar

[14]

G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052.  doi: 10.1088/0951-7715/12/4/318.  Google Scholar

[15]

W. T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, SpringerBriefs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-4487-9.  Google Scholar

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l'IHÉS, 51 (1980), 137–173.  Google Scholar

[17]

J. P. Kelly and T. Tennant, Topological entropy of set-valued functions, Houston J. Math., 43 (2017), 263-282.   Google Scholar

[18]

J. Kennedy and V. Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Theory Dynam. Systems, 38 (2018), 1499-1524.  doi: 10.1017/etds.2016.73.  Google Scholar

[19]

D. A. Lind, The entropies of topological markov shifts and a related class of algebraic integers, Ergodic Theory and Dynamical Systems, 4 (1984), 283-300.  doi: 10.1017/S0143385700002443.  Google Scholar

[20] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge university press, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[21]

R. McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 41 (1992), 1165-1209.  doi: 10.1512/iumj.1992.41.41058.  Google Scholar

[22]

W. Miller and E. Akin, Invariant measures for set-valued dynamical systems, Trans. Amer. Math. Soc., 351 (1999), 1203-1225.  doi: 10.1090/S0002-9947-99-02424-1.  Google Scholar

[23]

S. Pelikan, Invariant densities for random maps of the interval, Transactions of the American Mathematical Society, 281 (1984), 813-825.  doi: 10.1090/S0002-9947-1984-0722776-1.  Google Scholar

[24]

P. Walters, An Introduction to Ergodic Theory, volume 79., Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, volume 1 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1993. doi: 10.1090/gsm/001.  Google Scholar

[2]

L. Alvin and J. P. Kelly, Topological entropy of Markov set-valued functions, to appear in Ergodic Theory and Dynamical Systems.  Google Scholar

[3]

L. Alvin and J. P. Kelly, Markov set-valued functions and their inverse limits, Topology Appl., 241 (2018), 102-114.  doi: 10.1016/j.topol.2018.03.035.  Google Scholar

[4]

W. BahsounC. Bose and A. Quas, Deterministic representation for position-dependent random maps, Discrete & Continuous Dynamical Systems - A, 22 (2008), 529-540.  doi: 10.3934/dcds.2008.22.529.  Google Scholar

[5]

I. Banič and T. Lunder, Inverse limits with generalized Markov interval functions, Bull. Malays. Math. Sci. Soc., 39 (2016), 839-848.  doi: 10.1007/s40840-015-0187-0.  Google Scholar

[6]

I. Banič and M. črepnjak, Markov pairs, quasi Markov functions and inverse limits, Houston J. Math., 44 (2018), 695-707.   Google Scholar

[7]

R. Bowen, Invariant measures for Markov maps of the interval, Comm. Math. Phys., 69 (1979), 1-17.  doi: 10.1007/BF01941319.  Google Scholar

[8]

R. Bowen, Topological entropy for noncompact sets, Transactions of the American Mathematical Society, 184 (1973), 125-136.  doi: 10.1090/S0002-9947-1973-0338317-X.  Google Scholar

[9]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125-161.  doi: 10.1007/BF02773637.  Google Scholar

[10]

J. Buzzi, Exponential decay of correlations for random lasota–yorke maps, Communications in mathematical physics, 208 (1999), 25-54.  doi: 10.1007/s002200050746.  Google Scholar

[11]

W. Cordeiro and M. J. Pacífico, Continuum-wise expansiveness and specification for set-valued functions and topological entropy, Proc. Amer. Math. Soc., 144 (2016), 4261-4271.  doi: 10.1090/proc/13168.  Google Scholar

[12]

M. črepnjak and T. Lunder, Inverse limits with countably Markov interval functions, Glas. Mat. Ser. III, 51 (2016), 491-501.  doi: 10.3336/gm.51.2.14.  Google Scholar

[13]

G. Erceg and J. Kennedy, Topological entropy on closed sets in $[0, 1]^2$, Topology Appl., 246 (2018), 106-136.  doi: 10.1016/j.topol.2018.06.015.  Google Scholar

[14]

G. Froyland, Ulam's method for random interval maps, Nonlinearity, 12 (1999), 1029-1052.  doi: 10.1088/0951-7715/12/4/318.  Google Scholar

[15]

W. T. Ingram, An Introduction to Inverse Limits with Set-Valued Functions, SpringerBriefs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-4487-9.  Google Scholar

[16]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Publications Mathématiques de l'IHÉS, 51 (1980), 137–173.  Google Scholar

[17]

J. P. Kelly and T. Tennant, Topological entropy of set-valued functions, Houston J. Math., 43 (2017), 263-282.   Google Scholar

[18]

J. Kennedy and V. Nall, Dynamical properties of shift maps on inverse limits with a set valued function, Ergodic Theory Dynam. Systems, 38 (2018), 1499-1524.  doi: 10.1017/etds.2016.73.  Google Scholar

[19]

D. A. Lind, The entropies of topological markov shifts and a related class of algebraic integers, Ergodic Theory and Dynamical Systems, 4 (1984), 283-300.  doi: 10.1017/S0143385700002443.  Google Scholar

[20] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge university press, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar
[21]

R. McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 41 (1992), 1165-1209.  doi: 10.1512/iumj.1992.41.41058.  Google Scholar

[22]

W. Miller and E. Akin, Invariant measures for set-valued dynamical systems, Trans. Amer. Math. Soc., 351 (1999), 1203-1225.  doi: 10.1090/S0002-9947-99-02424-1.  Google Scholar

[23]

S. Pelikan, Invariant densities for random maps of the interval, Transactions of the American Mathematical Society, 281 (1984), 813-825.  doi: 10.1090/S0002-9947-1984-0722776-1.  Google Scholar

[24]

P. Walters, An Introduction to Ergodic Theory, volume 79., Springer-Verlag, New York-Berlin, 1982.  Google Scholar

Figure 1.  The graph of a Markov multi-map and its corresponding adjacency matrix
Figure 2.  Markov multi-map from Example 9.1
Figure 3.  Markov multi-maps from Example 10.2 (left) and Example 10.3 (right)
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