On the asymptotic relation of topological amenable group actions
DOI:
https://doi.org/10.12775/TMNA.2015.086Keywords
Topological G-action, entropy, asymptotic relation, generalized Pinsker formula, Pinsker sigma-algebraAbstract
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 $\A (\Phi)$ and we show that $\A (\Phi)$ 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 $\A (\Phi)$ is dense. These results are generalizations of those of Blanchard, Host and Ruette (\cite{BHR}) that concern the asymptotic relation for $\Z$-actions. We give an example of an expansive $G$-action ($G=\Z^2$) with $\A (\Phi)$ trivial which shows that the Bryant--Walters classical result (\cite{BW}) fails to be true in general case.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. Kaminski, A. Siemaszko and J. Szymanski, On deterministic and Kolmogorov extensions for topological ows, 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, Folner'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.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0