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Co-induction in dynamical systems

Published online by Cambridge University Press:  24 May 2011

ANTHONY H. DOOLEY  and
GUOHUA ZHANG
ANTHONY H. DOOLEY
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au)
GUOHUA ZHANG
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia (email: a.dooley@unsw.edu.au) School of Mathematical Sciences and LMNS, Fudan University, Shanghai 200433, PR China (email: chiaths.zhang@gmail.com)
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Abstract

If a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 3 , June 2012 , pp. 919 - 940
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Blanchard, F.. Fully positive topological entropy and topological mixing. Symbolic Dynamics and its Applications (Contemporary Mathematics, 135). American Mathematical Society, Providence, RI, 1992, pp. 95–105.CrossRefGoogle Scholar
[2]Blanchard, F.. A disjointness theorem involving topological entropy. Bull. Soc. Math. France 121(4) (1993), 465–478.CrossRefGoogle Scholar
[3]Blanchard, F., Glasner, E. and Host, B.. A variation on the variational principle and applications to entropy pairs. Ergod. Th. & Dynam. Sys. 17(1) (1997), 29–43.CrossRefGoogle Scholar
[4]Blanchard, F., Host, B., Maass, A., Martinez, S. and Rudolph, D. J.. Entropy pairs for a measure. Ergod. Th. & Dynam. Sys. 15(4) (1995), 621–632.CrossRefGoogle Scholar
[5]Danilenko, A. I.. Entropy theory from the orbital point of view. Monatsh. Math. 134(2) (2001), 121–141.CrossRefGoogle Scholar
[6]Dooley, A. H. and Golodets, V. Ya.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196(1) (2002), 1–18.CrossRefGoogle Scholar
[7]Dooley, A. H., Golodets, V. Ya., Rudolph, D. J. and Sineĺshchikov, S. D.. Non-Bernoulli systems with completely positive entropy. Ergod. Th. & Dynam. Sys. 28(1) (2008), 87–124.CrossRefGoogle Scholar
[8]Feldman, J.. New K-automorphisms and a problem of Kakutani. Israel J. Math. 24(1) (1976), 16–38.CrossRefGoogle Scholar
[9]Glasner, E.. A simple characterization of the set of μ-entropy pairs and applications. Israel J. Math. 102 (1997), 13–27.CrossRefGoogle Scholar
[10]Glasner, E., Thouvenot, J.-P. and Weiss, B.. Entropy theory without a past. Ergod. Th. & Dynam. Sys. 20(5) (2000), 1355–1370.CrossRefGoogle Scholar
[11]Glasner, E. and Ye, X.. Local entropy theory. Ergod. Th. & Dynam. Sys. 29(2) (2009), 321–356.CrossRefGoogle Scholar
[12]Hoffman, C.. A K counterexample machine. Trans. Amer. Math. Soc. 351(10) (1999), 4263–4280.CrossRefGoogle Scholar
[13]Huang, W. and Ye, X.. A local variational relation and applications. Israel J. Math. 151 (2006), 237–279.CrossRefGoogle Scholar
[14]Huang, W., Ye, X. and Zhang, G. H.. A local variational principle for conditional entropy. Ergod. Th. & Dynam. Sys. 26(1) (2006), 219–245.CrossRefGoogle Scholar
[15]Huang, W., Ye, X. and Zhang, G. H.. Relative entropy tuples, relative U.P.E. and C.P.E. extensions. Israel J. Math. 158 (2007), 249–283.CrossRefGoogle Scholar
[16]Huang, W., Ye, X. and Zhang, G. H.. Local entropy theory for a countable discrete amenable group action. Preprint, 2006, arXiv:1005.1335v1.Google Scholar
[17]del Junco, A.. Bernoulli shifts induced by K-automorphisms. Adv. Math. 25(1) (1977), 35–41.CrossRefGoogle Scholar
[18]Kalikow, S. A.. T,T −1 transformation is not loosely Bernoulli. Ann. of Math. (2) 115(2) (1982), 393–409.CrossRefGoogle Scholar
[19]Ollagnier, J. M. and Pinchon, D.. The variational principle. Studia Math. 72(2) (1982), 151–159.CrossRefGoogle Scholar
[20]Ornstein, D. S.. An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv. Math. 10 (1973), 49–62.CrossRefGoogle Scholar
[21]Ornstein, D. S.. A K automorphism with no square root and Pinsker’s conjecture. Adv. Math. 10 (1973), 89–102.CrossRefGoogle Scholar
[22]Ornstein, D. S.. Ergodic Theory, Randomness, and Dynamical Systems (Yale Mathematical Monographs, 5). Yale University Press, New Haven, CT–London, 1974.Google Scholar
[23]Ornstein, D. S. and Shields, P. C.. An uncountable family of K-automorphisms. Adv. Math. 10 (1973), 63–88.CrossRefGoogle Scholar
[24]Ornstein, D. S. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1–141.CrossRefGoogle Scholar
[25]Rudolph, D. J.. Two non-isomorphic K-automorphisms all of whose powers beyond one are isomorphic. Israel J. Math. 27(3–4) (1977), 277–298.CrossRefGoogle Scholar
[26]Rudolph, D. J. and Weiss, B.. Entropy and mixing for amenable group actions. Ann. of Math. (2) 151(3) (2000), 1119–1150.CrossRefGoogle Scholar
[27]Stepin, A. M. and Tagi-Zade, A. T.. Variational characterization of topological pressure of the amenable groups of transformations. Dokl. Akad. Nauk SSSR 254(3) (1980), 545–549 (in Russian).Google Scholar
[28]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York–Berlin, 1982.CrossRefGoogle Scholar
[29]Ward, T. and Zhang, Q.. The Abramov–Rokhlin entropy addition formula for amenable group actions. Monatsh. Math. 114(3–4) (1992), 317–329.CrossRefGoogle Scholar