Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T03:14:44.706Z Has data issue: false hasContentIssue false
  • English
  • Français

A minimal distal map on the torus with sub-exponential measure complexity

Published online by Cambridge University Press:  10 August 2018

WEN HUANG ,
LEIYE XU  and
XIANGDONG YE
WEN HUANG
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email [email protected], [email protected], [email protected]
LEIYE XU
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email [email protected], [email protected], [email protected]
XIANGDONG YE
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China email [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper the notion of sub-exponential measure complexity for an invariant Borel probability measure of a topological dynamical system is introduced. Then a minimal distal skew product map on the torus with sub-exponential measure complexity is constructed.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 4 , April 2020 , pp. 953 - 974
Copyright
© Cambridge University Press, 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ferenczi, S.. Measure-theoretic complexity of ergodic systems. Israel J. Math. 100 (1997), 189–207.Google Scholar
Furstenberg, H.. The structure of distal flows. Amer. J. Math. 85 (1963), 477–515.Google Scholar
Glasner, E.. Ergodic Theory via Joinning (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.Google Scholar
Huang, W., Li, J., Thouvenot, J., Xu, L. and Ye, X.. Bounded complexity, mean equicontinuity and discrete spectrum. Preprint, 2018, arXiv:1806.02980.Google Scholar
Huang, W., Wang, Z. and Ye, X.. Measure complexity and Möbius disjointness. Preprint, 2017, arXiv:1707.06345.Google Scholar
Kac, M.. On the notion of recurrence in discrete stochastic processes. Bull. Amer. Math. Soc. 53 (1947), 1002–1010.Google Scholar
Katok, A.. Lyapunov exponents, entropy and the periodic orbits for diffeomorphisms. Publ. Math. Inst. Hautes Études Sci. 51 (1980), 137–173.Google Scholar
Kočergin, A.V.. The homology of functions over dynamical systems. Dokl. Akad. Nauk SSSR 231(4) (1976), 795–798 (Russian).Google Scholar
Lindenstrauss, E.. Measurable distal and topological distal systems. Ergod. Th. & Dynam. Sys. 19(4) (1999), 1063–1076.Google Scholar
Lindenstrauss, E. and Tsukamoto, M.. From rate distortion theory to metric mean dimension: variational principle. IEEE Trans. Inform. Theory 64(5) (2018), 3590–3609.Google Scholar
Parry, W.. Zero entropy of distal and related transformations. Topological Dynamics (Symposium, Colorado State University, Ft. Collins, CO). Benjamin, New York, 1967, pp. 383–389.Google Scholar
Qiao, Y.. Topological complexity, minimality and systems of order two on torus. Sci. China Math. 59 (2016), 503–514.Google Scholar
Sacker, R.J. and Sell, G.R.. Finite extensions of minimal transformation groups. Trans. Amer. Math. Soc. 190 (1974), 325–334.Google Scholar
Velozo, A. and Velozo, R.. Rate distortion theory, metric mean dimension and measure theoretic entropy. Preprint, 2017, arXiv:1707.05762.Google Scholar