November  2022, 27(11): 6589-6604. doi: 10.3934/dcdsb.2022010

Metric entropy for set-valued maps

Departamento de Matemáticas, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile

*Corresponding author: Kendry J. Vivas

Received  October 2021 Revised  December 2021 Published  November 2022 Early access  January 2022

In this article we define a notion of metric entropy for an invariant measure associated to a set-valued map $ F $ on a compact metric space $ X $. Besides, we describe its main properties and prove the Half Variational Principle, which relates the metric entropy with the notion of topological entropy given in [13] for this class of maps.

Citation: Kendry J. Vivas, Víctor F. Sirvent. Metric entropy for set-valued maps. Discrete and Continuous Dynamical Systems - B, 2022, 27 (11) : 6589-6604. doi: 10.3934/dcdsb.2022010
References:
[1]

R. L. AdlerA. G. Konhein and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Reprint of the 1990 edition edition, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[3]

J.-P. AubinH. Frankowska and A. Lazota, Poincaré's recurrence theorem for set-valued dynamical systems, Ann. Polon. Math., 54 (1991), 85-91.  doi: 10.4064/ap-54-1-85-91.

[4]

L. Boltzmann, Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen uber das Wärmegleichgewicht, Wiener Berichte., 76 (1877), 373-435. 

[5]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[6]

D. Carrasco-OliveraR. Metzger Alvan and C. A. Morales Rojas, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.

[7]

D. Carrasco-OliveraR. Metzger and C. A. Morales, Logarithmic expansion, entropy, and dimension for set-valued maps, Optimal Control, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 178 (2020), 31-40. 

[8]

J. Casimiro, Metric entropy and topological Entropy: The Variational Principle, Master thesis in Mathematics and Applications, Tecnico Lisboa, 2014.

[9]

R. Clausius, Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Annalen der Physik, 125 (1865), 353-400.  doi: 10.1002/andp.18652010702.

[10]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems, Mathematics of the USSR-Izvestiya, 5 (1971), 337-378.  doi: 10.1070/IM1971v005n02ABEH001050.

[11]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bulletin of the London Mathematical Society, 3 (1971), 176-180.  doi: 10.1112/blms/3.2.176.

[12]

A. Y. Hin$\check{\mathrm{v}}$in, On the basic theorems of information theory, Uspehi Mat. Nauk, 11 (1956), 17-75. 

[13]

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

[14]

X. F. LuoX. X. Nie and J. D. Yin, On the shadowing property and shadowable point of set-valued dynamical systems, Acta Mathematica Sinica, English series, 12 (2020), 1384-1394.  doi: 10.1007/s10114-020-9331-3.

[15]

B. McMillan, The basic theorems of information theory, Ann. Math. Statistics, 24 (1953), 196-219.  doi: 10.1214/aoms/1177729028.

[16]

R. MetzgerC. A. Morales Rojas and P. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975.  doi: 10.3934/dcdsb.2017115.

[17]

M. J. Pacifico and J. Vieitez, Expansiveness, Lyapunov exponents and entropy for set valued maps, arXiv: 1709.05739v3.

[18]

B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, Houston J. Math., 44 (2018), 665-677. 

[19]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 623-656.  doi: 10.1002/j.1538-7305.1948.tb01338.x.

[20]

J. Sinai, On the concept of entropy for a dynamic system (Russian), Dokl. Akad. Nauk SSSR, 124 (1959), 768-771. 

[21] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics 151, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316422601.
[22]

J. von Newmann, Mathematische Grundlagen der Quantenmechanik, Unveränderter Nachdruck der ersten Auflage von 1932, Die Grundlehren der mathematischen Wissenschaften, Band 38, Springer-Verlag, Berlin-New York, 1968.

show all references

References:
[1]

R. L. AdlerA. G. Konhein and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Reprint of the 1990 edition edition, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.

[3]

J.-P. AubinH. Frankowska and A. Lazota, Poincaré's recurrence theorem for set-valued dynamical systems, Ann. Polon. Math., 54 (1991), 85-91.  doi: 10.4064/ap-54-1-85-91.

[4]

L. Boltzmann, Über die beziehung dem zweiten Haubtsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen uber das Wärmegleichgewicht, Wiener Berichte., 76 (1877), 373-435. 

[5]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[6]

D. Carrasco-OliveraR. Metzger Alvan and C. A. Morales Rojas, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B., 20 (2015), 3461-3474.  doi: 10.3934/dcdsb.2015.20.3461.

[7]

D. Carrasco-OliveraR. Metzger and C. A. Morales, Logarithmic expansion, entropy, and dimension for set-valued maps, Optimal Control, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 178 (2020), 31-40. 

[8]

J. Casimiro, Metric entropy and topological Entropy: The Variational Principle, Master thesis in Mathematics and Applications, Tecnico Lisboa, 2014.

[9]

R. Clausius, Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie, Annalen der Physik, 125 (1865), 353-400.  doi: 10.1002/andp.18652010702.

[10]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems, Mathematics of the USSR-Izvestiya, 5 (1971), 337-378.  doi: 10.1070/IM1971v005n02ABEH001050.

[11]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bulletin of the London Mathematical Society, 3 (1971), 176-180.  doi: 10.1112/blms/3.2.176.

[12]

A. Y. Hin$\check{\mathrm{v}}$in, On the basic theorems of information theory, Uspehi Mat. Nauk, 11 (1956), 17-75. 

[13]

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

[14]

X. F. LuoX. X. Nie and J. D. Yin, On the shadowing property and shadowable point of set-valued dynamical systems, Acta Mathematica Sinica, English series, 12 (2020), 1384-1394.  doi: 10.1007/s10114-020-9331-3.

[15]

B. McMillan, The basic theorems of information theory, Ann. Math. Statistics, 24 (1953), 196-219.  doi: 10.1214/aoms/1177729028.

[16]

R. MetzgerC. A. Morales Rojas and P. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975.  doi: 10.3934/dcdsb.2017115.

[17]

M. J. Pacifico and J. Vieitez, Expansiveness, Lyapunov exponents and entropy for set valued maps, arXiv: 1709.05739v3.

[18]

B. E. Raines and T. Tennant, The specification property on a set-valued map and its inverse limit, Houston J. Math., 44 (2018), 665-677. 

[19]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J., 27 (1948), 623-656.  doi: 10.1002/j.1538-7305.1948.tb01338.x.

[20]

J. Sinai, On the concept of entropy for a dynamic system (Russian), Dokl. Akad. Nauk SSSR, 124 (1959), 768-771. 

[21] M. Viana and K. Oliveira, Foundations of Ergodic Theory, Cambridge Studies in Advanced Mathematics 151, Cambridge University Press, Cambridge, 2016.  doi: 10.1017/CBO9781316422601.
[22]

J. von Newmann, Mathematische Grundlagen der Quantenmechanik, Unveränderter Nachdruck der ersten Auflage von 1932, Die Grundlehren der mathematischen Wissenschaften, Band 38, Springer-Verlag, Berlin-New York, 1968.

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