## Topological Methods in Nonlinear Analysis

### On the asymptotic relation of topological amenable group actions

#### Abstract

For a topological action $\Phi$ of a countable amenable orderable group $G$ on a compact metric space we introduce a concept of the asymptotic relation $\mathbf{A} (\Phi)$ and we show that $\mathbf{A} (\Phi)$ is non-trivial if the topological entropy $h(\Phi)$ is positive. It is also proved that if the Pinsker $\sigma$-algebra $\pi_{\mu}(\Phi)$ is trivial, where $\mu$ is an invariant measure with full support, then $\mathbf{A} (\Phi)$ is dense. These results are generalizations of those of Blanchard, Host and Ruette ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) that concern the asymptotic relation for $\mathbb{Z}$-actions.

We give an example of an expansive $G$-action ($G=\mathbb{Z}^2$) with $\mathbf{A} (\Phi)$ trivial which shows that the Bryant-Walters classical result ([B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60-66]) fails to be true in general case.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 47, Number 1 (2016), 43-54.

Dates
First available in Project Euclid: 23 March 2016

https://projecteuclid.org/euclid.tmna/1458740728

Digital Object Identifier
doi:10.12775/TMNA.2015.086

Mathematical Reviews number (MathSciNet)
MR3469046

Zentralblatt MATH identifier
1362.37025

#### Citation

Bułatek, Wojciech; Kamiński, Brunon; Szymański, Jerzy. On the asymptotic relation of topological amenable group actions. Topol. Methods Nonlinear Anal. 47 (2016), no. 1, 43--54. doi:10.12775/TMNA.2015.086. https://projecteuclid.org/euclid.tmna/1458740728

#### References

• P. Billingsley, Ergodic Theory and Information, Wiley, New York 1965.
• F. Blanchard, B. Host and S. Ruette, Asymptotic pairs in positive entropy systems, Ergodic Theory Dynam. Systems 22 (2002), 671–686.
• B.F. Bryant and P. Walters, Asymptotic properties of expansive homeomorphisms, Math. System Theory 3 (1969), 60–66.
• M. Denker, Ch. Grillenberger and K. Sigmund, Ergodic theory on compact spaces, Lectures Notes Math. 527, Springer Verlag, Berlin, Heidelberg, New York, 1976.
• T. Downarowicz, Entropy in dynamical systems, Cambridge University Press, New Math. Monogr. 18 Cambridge, New York, Melbourne (2011).
• T. Downarowicz and Y. Lacroix, Topological entropy zero and asymptotic pairs, Israel J. Math. 189 (2012), 323–336.
• L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London (1963).
• F.P. Greenleaf, Ergodic theorems and the construction of summing sequences in amenable locally compact groups, Comm. Pure Appl. Math. 26 (1973), 29–46.
• W. Huang and X. Ye, Devaney's chaos or $2$-scattering implies Li–York's chaos, Topology and its applications 117 (2002), 259–272.
• B. Kamiński, A. Siemaszko and J. Szymański, On deterministic and Kolmogorov extensions for topological flows, Topol. Methods Nonlinear Anal. 31 (2008), 191–204.
• J.C. Kieffer, A generalized Shannon–McMillan theorem for the action of an amenable group on a probability space, Ann. Probability 3 (1975), 1031–1037.
• N.F.G. Martin and J.W. England, Mathematical theory of entropy, Encyclopedia of Mathematics and its Applications 12, Addison–Wesley Publishing Co., Reading, Mass. 1981.
• W. Mlak, Hilbert spaces and operator theory, PWN – Polish Scientific Publishers, Warszawa, Kluwer Academic Publishers, Dodrecht, 1991.
• I. Namioka, Følner's conditions for amenable semi-groups, Math. Scand. 15 (1964), 18–28.
• J.M. Ollagnier and D. Pinchon, The variational principle, Studia Math. 72 (1982), 151–159.
• V.A. Rokhlin, On the fundamental ideas of measure theory, Mat. Sb. 25 (67) (1949), 107–150.
• A. V. Safonov, Informational pasts in groups, Izv. Akad. Nauk. SSSR 47 (1983), 421–426.
• A.M. Stepin and A.T. Tagi-Zade, Variational characterization of topological pressure of the amenable groups of transformations, Dokl. Akad. Nauk SSSR 254 (1980), 545–549.