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 and 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.

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

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

[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.

[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.

[24]

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

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.

[2]

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

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[16]

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

[17]

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

[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.

[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.

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

[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.

[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.

[24]

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

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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