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Slow entropy and differentiable models for infinite-measure preserving ℤk actions

Published online by Cambridge University Press:  17 January 2012

MICHAEL HOCHMAN
MICHAEL HOCHMAN*
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Rd, Princeton, NJ 08544, USA (email: [email protected])
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Abstract

We define ‘slow’ entropy invariants for ℤd actions on infinite measure spaces, which measure growth of itineraries at subexponential scales. We use this notion to construct infinite-measure preserving ℤ2 actions which cannot be realized as a group of diffeomorphisms of a compact manifold preserving a Borel measure, in contrast to the situation for ℤ actions, where every infinite-measure preserving action can be realized in this way.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 653 - 674
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Aaronson, J.. An Introduction to Infinite Ergodic Theory (Mathematical Surveys and Monographs, 50). American Mathematical Society, Providence, RI, 1997.CrossRefGoogle Scholar
[2]Avni, N.. Entropy theory for cross-sections. Geom. Funct. Anal. 19(6) (2010), 1515–1538.CrossRefGoogle Scholar
[3]Chacon, R. V. and Ornstein, D. S.. A general ergodic theorem. Illinois J. Math. 4 (1960), 153–160.CrossRefGoogle Scholar
[4]Danilenko, A. and Silva, C.. Ergodic theory: nonsingular transformations. Encyclopedia of Complexity and Systems Science, Part 5. Springer, 2009, pp. 3055–3083.CrossRefGoogle Scholar
[5]Danilenko, A. I. and Rudolph, D. J.. Conditional entropy theory in infinite measure and a question of Krengel. Israel J. Math. 172 (2009), 93–117.CrossRefGoogle Scholar
[6]Dou, D., Huang, W. and Park, K. K.. Entropy dimension of topological dynamical systems. Trans. Amer. Math. Soc. 363(2) (2011), 659–680.CrossRefGoogle Scholar
[7]Furstenberg, H. and Weiss, B.. The finite multipliers of infinite ergodic transformations. The Structure of Attractors in Dynamical Systems (Proc. Conf. North Dakota State Univ., Fargo, ND, 1977) (Lecture Notes in Mathematics, 668). Springer, Berlin, 1978, pp. 127–132.CrossRefGoogle Scholar
[8]Galatolo, S., Kim, D. H. and Park, K. K.. The recurrence time for ergodic systems with infinite invariant measures. Nonlinearity 19(11) (2006), 2567–2580.CrossRefGoogle Scholar
[9]Hochman, M.. A ratio ergodic theorem for multiparameter non-singular actions. J. Eur. Math. Soc. 12(2) (2010), 365–383.CrossRefGoogle Scholar
[10]Janvresse, É. and de la Rue, T.. Zero Krengel entropy does not kill Poisson entropy. Ann. Henri Poincaré to appear, Preprint, 2010, http://arxiv.org/abs/0910.2566.Google Scholar
[11]Janvresse, É., Meyerovitch, T., Roy, E. and de la Rue, T.. Poisson suspensions and entropy for infinite transformations. Trans. Amer. Math. Soc. 362(6) (2010), 3069–3094.CrossRefGoogle Scholar
[12]Katok, A. and Thouvenot, J.-P.. Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. Ann. Inst. Henri. Poincaré Probab. Stat. 33(3) (1997), 323–338.CrossRefGoogle Scholar
[13]Krengel, U.. Entropy of conservative transformations. Z. Wahr. Verw. Gebiete 7 (1967), 161–181.CrossRefGoogle Scholar
[14]Krengel, U.. Transformations without finite invariant measure have finite strong generators. Contributions to Ergodic Theory and Probability (Proc. Conf. Ohio State Univ., Columbus, OH, 1970). Springer, Berlin, 1970, pp. 133–157.CrossRefGoogle Scholar
[15]Krieger, W.. On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149 (1970), 453–464.CrossRefGoogle Scholar
[16]Parry, W.. Entropy and Generators in Ergodic Theory. W. A. Benjamin, Inc., New York–Amsterdam, 1969.Google Scholar
[17]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar