اندازه قطری و آنتروپی دستگاه‌های دینامیکی

نوع مقاله : مقاله پژوهشی

نویسندگان

گروه ریاضی، دانشکده علوم پایه، دانشگاه قم، قم، ایران

چکیده

در این مقاله، ابتدا مفهوم اندازۀ قطری متناظر با یک اندازۀ پایا تحت یک دستگاه دینامیکی فشرده را تعریف کرده و به بررسی برخی از خواص آن می‌پردازیم. در‌ نهایت، نشان می‌دهیم که انتگرال‌گیری از یک تابع مناسب، نسبت به اندازۀ قطری، منجر به آنتروپی متریک یک دستگاه دینامیکی فشرده می‌شود.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Diagonal measure and entropy of dynamical systems

نویسندگان [English]

  • Mehdi Rahimi
  • Nahid Bidabadi
Department of Mathematics, University of Qom, Qom, Iran
چکیده [English]

In this paper, we first define the concept of diagonal measure corresponding to an invariant measure of a compact dynamical system and will investigate some of its properties. Finally, we show that, integrating a suitable function with respect to the diagonal measure results in the metric entropy of a compact dynamical system.

کلیدواژه‌ها [English]

  • Metric entropy
  • Information function
  • Invariant measure
  • Ergodic measure
  • Diagonal measure
[1] Adler, R.L., Konheim, A.G., & McAndrew, M.H. (1965). Topological entropy. Trans. Amer. Math. Soc, 114, 309–319. DOI: https://doi.org/10.1090/S0002-9947-1965-0175106-9.
[2] Bowen, R. (1976). Invariant measures for Markov maps of the interval. Comm. Math. Physics, 69, 1–17. DOI: https://doi.org/10.1007/BF01941319.
[3] Breiman, L. (1957). The individual theorem of information theory. Ann of Math Stat, 28, 809–811; errata, 31 (1960), 809–810. DOI: https://doi.org/10.1214/aoms/1177706899.
[4] Brin, M., & Katok, A. (1983). On local entropy in geometric dynamics. 30–38, New York, SpringerVerlag, (Lecture Notes in Mathematics 1007). DOI: https://doi.org/10.1007/bfb0061408.
[5] Dinaburg, E.I. (1970). The relation between topological entropy and metric entropy. Soviet Math, 11, 13–16.
[6] Kolmogorov, A.N. (1958). New metric invariant of transitive dynamical systems and endomorphisms of Lebesgue spaces. Doklady of Russian Academy of Sciences, 119, 861–864.
[7] Mañé, R. (1987). Ergodic theory and differentiable dynamics. Springer-Verlag, Berlin, Heidelberg, New York. DOI: https://doi.org/10.1007/978-3-642-70335-5.
[8] McMillan, B. (1953). The basic theorems of information theory. Ann. of Math. Statistics, 24, 196–219. DOI: https://doi.org/10.1214/aoms/1177729028.
[9] Pesin, Ya. (1977). Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys, 32, 54–114. DOI: https://doi.org/10.1070/RM1977v032n04ABEH001639.
[10] Rahimi, M. (2021). A Spectral Representation for the Entropy of Topological Dynamical Systems. J Dyn Control Syst, 27, 573–584. DOI: https://doi.org/10.1007/s10883-020-09519-w.
[11] Rahimi, M., & Riazi, A. (2012). Entropy operator for continuous dynamical systems of finite topological entropy. Bulletin of the Iranian Mathematical Society, 38, 883–892.
[12] Rahimi, M., & Riazi, A. (2012). Entropy functional for continuous systems of finite entropy. Acta Mathematica Scientia, 32B, 775–782. DOI: https://doi.org/10.1016/S0252-9602(12)60057-5.
[13] Ruelle, D. (1987). An inequality for the entropy of differential maps. Bol. Soc. Bras. de Mat, 9, 83–87. DOI: https://doi.org/10.1007/bf02584795.
[14] Shannon, C. (1948). A mathematical theory of communication. Bell Syst. Tech. Journal, 27, 379–423. DOI: https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
[15] Sinai, Ya.G. (1959). On the notion of entropy of a dynamical system. Doklady of Russian Academy of Sciences, 124, 768–771. DOI: https://doi.org/10.1007/978-0-387-87870-6_1.
[16] Walters, P. (1982). An introduction to ergodic theory. Springer-Verlag. DOI: https://doi.org/10.1007/springerreference_60354