Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-01-09T18:04:28.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Directional Kronecker algebra for $\mathbb {Z}^q$-actions

Part of: Ergodic theory

Published online by Cambridge University Press:  28 January 2022

CHUNLIN LIU  and
LEIYE XU
CHUNLIN LIU*
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (e-mail: [email protected])
LEIYE XU
Affiliation:
CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China (e-mail: [email protected])
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, directional sequence entropy and directional Kronecker algebra for $\mathbb {Z}^q$ -systems are introduced. The relation between sequence entropy and directional sequence entropy are established. Meanwhile, directional discrete spectrum systems and directional null systems are defined. It is shown that a $\mathbb {Z}^q$ -system has directional discrete spectrum if and only if it is directional null. Moreover, it turns out that a $\mathbb {Z}^q$ -system has directional discrete spectrum along q linearly independent directions if and only if it has discrete spectrum.

Keywords

directional sequence entropydirectional Kronecker algebradirectional discrete spectrum system

MSC classification

Primary: 37A25: Ergodicity, mixing, rates of mixing
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37A05: Measure-preserving transformations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 4 , April 2023 , pp. 1324 - 1350
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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

Akin, H.. On the directional entropy of ${\mathbb{Z}}^2$ -actions generated by additive cellular automata. Appl. Math. Comput. 170 (2005), 339–346.Google Scholar
Broderick, R., Cyr, V. and Kra, B.. Complexity and directional entropy in two dimensions. Israel J. Math. 215 (2016), 135–162.CrossRefGoogle Scholar
Courbage, M. and Kaminski, B.. On the directional entropy of ${\mathbb{Z}}^2$ -actions generated by cellular automata. Studia Math. 153 (2002), 285–295.10.4064/sm153-3-5CrossRefGoogle Scholar
Dou, D., Huang, W. and Park, K. K.. Entropy dimension of measure preserving systems. Trans. Amer. Math. Soc. 371 (2019), 7029–7065.CrossRefGoogle Scholar
Downarowicz, T.. Entropy in Dynamical Systems (New Mathematical Monographs, 18). Cambridge University Press, Cambridge, 2011.10.1017/CBO9780511976155CrossRefGoogle Scholar
Durrett, R.. Probability Theory and Examples, 4th edn. Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511779398CrossRefGoogle Scholar
Einsiedler, M. and Ward, T.. Ergodic Theory with a View Towards Number Theory (Graduate Text in Mathematics, 259). Springer, London, 2011.10.1007/978-0-85729-021-2CrossRefGoogle Scholar
Folland, G. B.. Real Analysis: Modern Techniques and Their Applications. John Wiley & Sons, Inc., New Jersey, 1999.Google Scholar
Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys and Monographs, 101). American Mathematical Society, Providence, RI, 2003.10.1090/surv/101CrossRefGoogle Scholar
Huang, W., Maass, A. and Ye, X.. Sequence entropy pairs and complexity pairs for a measure. Ann. Inst. Fourier (Grenoble) 54 (2004), 1005–1028.CrossRefGoogle Scholar
Hulse, P.. On the sequence entropy of transformations with quasidiscrete spectrum. J. Lond. Math. Soc. (2) 20 (1979), 128–136.10.1112/jlms/s2-20.1.128CrossRefGoogle Scholar
Katok, A. B. and Thouvenot, J. P.. Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. Ann. Inst. Henri Poincaré Probab. Stat. 33 (1997), 323–338.10.1016/S0246-0203(97)80094-5CrossRefGoogle Scholar
Kerr, D. and Li, H.. Combinatorial independence in measurable dynamics. J. Funct. Anal. 256 (2009) 1341–1386.CrossRefGoogle Scholar
Kolmogorov, A. N.. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR 119 (1958), 861–864 (in Russian).Google Scholar
Koopman, B. O. and von Neumann, J.. Dynamical systems of continuous spectra. Proc. Natl Acad. Sci. USA 18 (1932), 255–263.CrossRefGoogle ScholarPubMed
Kušhnirenko, A. G.. Metric invariants of entropy type. Uspekhi Mat. Nauk 22 (1967), 57–65 (in Russian).Google Scholar
Milnor, J.. On the entropy geometry of cellular automata. Complex Syst. 2 (1988), 357–386.Google Scholar
Muscat, J.. Functional Analysis: An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras. Springer, Cham, 2014.CrossRefGoogle Scholar
Park, K. K.. Continuity of directional entropy. Osaka J. Math. 31 (1994), 613–628.Google Scholar
Park, K. K.. On directional entropy functions. Israel J. Math. 113 (1999), 243–267.CrossRefGoogle Scholar
Saleski, A.. Sequence entropy and mixing. J. Math. Anal. Appl. 60 (1977), 58–66.10.1016/0022-247X(77)90047-6CrossRefGoogle Scholar
Stein, E. M. and Shakarchi, R.. Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, Princeton, NJ, 2005.CrossRefGoogle Scholar
Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematic, 79). Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
Zhu, Y.. A note on two types of Lyapunov exponents and entropies for ${\mathbb{Z}}^k$ -actions. J. Math. Anal. Appl. 461 (2018), 38–50.CrossRefGoogle Scholar
Zimmer, J. R.. Extensions of ergodic group actions. Illinois J. Math. 20 (1976), 373–409.CrossRefGoogle Scholar