Directional Entropy and Pinsker $\sigma $-Algebra for $\mathbb {Z}^{2}$-Actions

Abstract

In this paper, we study the directional entropy along irrational directions in $\mathbb {R}^2$ and present the structure of directional Pinsker $\sigma $-algebra of $\mathbb {Z}^2$-MPSs.

Keywords

Directional Pinsker $\sigma $-algebra / Directional entropy / 28D20 / 37A35

References

[1] Adler RL, Konheim AG, McAndrew MH. Topological entropy. Trans. Am. Math. Soc., 1965, 114: 309-319

[2] Broderick R, Cyr V, Kra B. Complexity and directional entropy in two dimensions. Israel J. Math., 2016, 215: 135-162

[3] Boyle M, Lind D. Expansive subdynamics. Trans. Am. Math. Soc., 1997, 349: 55-102

[4] Dooley A, Zhang G. Local entropy theory of a random dynamical system. Mem. Am. Math. Soc., 2015, 233(1099): vi+106

[5] Einsiedler M, Lind D. Algebraic Zd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^d$$\end{document}-actions on entropy rank one. Trans. Am. Math. Soc., 2004, 356: 1831-1977

[6] Einsiedler M, Ward T. Ergodic Theory with a View Towards Number Theory, 2011, London, Springerxviii+481

[7] Huang W, Ye X. A local variational relation and applications. Israel J. Math., 2006, 151: 237-279

[8] Huang W, Ye X, Zhang G. Relative entropy tuples, relative U.P.E. and C.P.E. extensions. Israel J. Math., 2007, 158: 249-283

[9] Huang W, Ye X, Zhang G. Local entropy theory for a countable discrete amenable group action. J. Funct. Anal., 2011, 261(4): 1028-1082

[10] Glasner, E., Weiss, B.: Topological entropy of extensions, Ergodic Theory and its connections with harmonic nnalysis (Alexandria, 1993), pp. 299–307, London Math. Soc. Lecture Note Ser., 205. Cambridge Univ. Press, Cambridge (1995)

[11] Kolmogorov AN. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. (Russian) Dokl. Akad. Nauk SSSR (N.S.), 1958, 119: 861-864

[12] Kolmogorov AN. Entropy per unit time as a metric invariant of automorphisms. (Russian) Dokl. Akad. Nauk SSSR, 1959, 124: 754-755

[13] Kaminski B, Park K. On the directional entropy for Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^2$$\end{document}-actions on a Lebesgue space. Studia Math., 1999, 133: 39-51

[14] Lemańczyk M, Siemaszko A. A note on the existence of a largest topological factor with zero entropy. Proc. Am. Math. Soc., 2001, 129(2): 475-482

[15] Milnor J. On the entropy geometry of cellular automata. Complex Syst., 1988, 2(3): 357-385

[16] Park K. Continuity of directional entropy for a class of Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^2$$\end{document}-actions. J. Korean Math. Soc., 1995, 32: 573-582

[17] Park K. On directional entropy functions. Israel J. Math., 1999, 113: 243-267

[18] Park K, Lee U. Entropy pairs of Z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^2$$\end{document} and their directional properties. Stud. Math., 2004, 3: 255-274

[19] Park K, Siemaszko A. Relative topological Pinsker factors and entropy pairs. Monatsh. Math., 2001, 134(1): 67-79

[20] Pinsker MS. Dynamical systems with completely positive or zero entropy. Dokl. Akad. Nauk SSSR, 1960, 133: 1025-1026

[21] Robinson E, Sahin A. Rank-one Zd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^d$$\end{document}-actions and directional entropy. Ergodic Theory Dyn. Syst., 2011, 31: 285-299

[22] Sinai, Ya.G.: An answer to a question by J. Milnor. Comment. Math. Helv. 60(2), 173–178 (1985)

[23] Sinai, Ya.G.: On the concept of entropy for a dynamic system. (Russian) Dokl. Akad. Nauk SSSR 124, 768–771 (1959)

[24] Wei, R., Xu, L., Zheng, L., Zhou, X.: Bounded directional complexity and rigidity for Zq\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}^{q}$$\end{document} -actions. In preparation

[25] Zhu Y. A note on two types of Lyapunov exponents and entropies for Zk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z} ^k$$\end{document}-actions. J. Math. Anal. Appl., 2018, 461: 38-50

[26] Zhang J, Zhang W. Directional entropy of Z+k\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Z}_+^k$$\end{document}-actions. Stoch. Dyn., 2016, 1611650004, 22

Funding

National Natural Science Foundation of China(12031019)

Fundamental Research Funds for the Central Universities(2020kfyXJJS036)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature