On induced average entropy of random dynamical systems

Document Type : Original Article

Authors

Department of Mathematics, University of Qom, Qom, Iran

Abstract

In this paper, we consider random dynamical systems as a family of continuous maps on a compact metric space, parameterized by a measure space via a noise map. We define a new type of induced entropy for such families, called induced average entropy and will establish the connection between this new quantity and the induced upper capacity topological entropy of its skew product and the topological entropy of the noise map. This relation resembles the Abramov-Rokhlin theorem in the topological setting.

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References

[1] Adler, R.L., Konheim, A.G., & McAndrew, M.H. (1965). Topological entropy. Trans. Amer. Math. Soc., 114, 309–319. DOI:
http://dx.doi.org/10.1090/S0002-9947-1965-0175106-9.
[2] Arnold, L. (2003). Random Dynamical Systems. Springer Monographs in Mathematics, Springer Berlin, Heidelberg. DOI: http://dx.doi.org/10.1007/978-3-662-12878-7.
[3] Bogenschutz, T. (1992). Entropy, pressure and a variational principle for random dynamical systems. Random Comput. Dynam., 1, 99–116.
[4] Bogenschutz, T. (1993). Equilibrium states for random dynamical systems. Ph.D. Thesis, Bremen University.
[5] Bogenschutz, T., & Crauel, H. (1992). The Abramov-Rokhlin formula. In: Krengel U., Richter K., Warstat V. (eds) Ergodic Theory and Related Topics III. Lecture Notes in Mathematics, vol 1514. Springer Berlin, Heidelberg. DOI: http://dx.doi.org/10.1007/BFb0097526.
[6] Jaerisch, J., Kessebhmer, M., & Lamei, S. (2014). Induced topological pressure for countable state Markov shifts. Stoch. Dyn., 14. DOI: http://dx.doi.org/10.1142/S0219493713500160.
[7] Katok, A. (1980). Lyapunov exponents, entropy and periodic points for diffeomorphisms. Publ. Math. IHÉS, 51, 137–173.
[8] Kifer, Y. (1986). Ergodic Theory of Random Transformations. Springer, Progress in Probability and Statistics, 10, 13–25. DOI: http://dx.doi.org/10.1007/978-1-4684-9175-3.
[9] Kolmogorov, A.N. (1958). New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces. Doklady of Russian Academy of Sciences, 119, N5, 861–864.
[10] Liu, P.D. (2001). Dynamics of random transformations: smooth ergodic theory. Ergodic Theory Dynam. Systems, 21,
1279–1319. DOI: http://dx.doi.org/10.1017/S0143385701001614.
[11] Rahimi, M., & Ghodrati, A. (2024). Average topological pressure and a variational principle. J. Dyn. Control Syst., 30. DOI: https://doi.org/10.1007/s10883-024-09688-y.
[12] Rahimi, M., Shakouri, A., & Anjedani, M.M. (2021). A note on local entropy of random dynamical systems. Mathematical Analysis and convex optimization, 2, 87–97.
[13] Savchenko, S.V. (1998). Special flows constructed from countable topological Markov chains. Funct Anal Its Appl., 32, 32–41. DOI: http://dx.doi.org/10.1007/BF02465754.
[14] Sinai, Ya.G. (1959). On the notion of entropy of a dynamical system. Doklady of Russian Academy of Sciences, 124, 768–771.
[15] Xing, Z., & Chen, E. (2015). Induced topological pressure for topological dynamical systems. J. Math. Phys., 56, 022707. DOI: http://dx.doi.org/10.1063/1.4908554.
[16] Xing, Z., Chen, E., & Yin, Z. (2018). Katok formula for the induced measure-theoretic entropy. Dyn. Syst., 33, 195–206. DOI: http://dx.doi.org/10.1080/14689367.2017.1329402.
[17] Yang, K., Chen, E., & Zhou, X. (2022). On the induced measure-theoretic entropy for random dynamical systems. Stoch. Dyn., 22, DOI: http://dx.doi.org/10.1142/S0219493722500307.
[18] Zhu, Y.J. (2008). On local entropy of random transformations. Stoch. Dyn., 8, 197–207. DOI: http://dx.doi.org/10.1142/S0219493708002275.
[19] Zhu, Y.J. (2009). Two notes on measure-theoretic entropy of random dynamical systems. Acta Math. Sin. (Engl. Ser.), 25, 961–970. DOI: http://dx.doi.org/10.1007/s10114-009-7206-8.
Measure Algebras and Applications
Volume 3, Issue 2 - Serial Number 5
December 2025
Pages 70-81

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