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Subadditive convergence via hyperfinite equivalence relations

Banff International Research Station for Mathematical Innovation and Discovery

I would like to give a talk about a new Ornstein-Weiss type subadditive convergence theorem along hyperfinite exhaustions of pmp Borel equivalence relations. In collaboration with Amos Nevo (Techion), we used this result to define a new notion of entropy (cocycle entropy) for pmp actions of abritrary countable groups. It has turned out that for free actions, cocycle entropy coincides with Rokhlin entropy which is studied by Brandon Seward et al. However, using subadditive convergence techniques in order to assign entropy values to measured partitions, our definition is a priori quite different and more in line with the classical Kolmogorov-Sinai approach. Moreover, extending Elon Lindenstrauss' techniques to hyperfinite equivalence relations, we were able to settle the underlying Shannon-McMillan-Breiman theorem for a vast collection of pmp actions of general countable groups. Being of very general nature, we expect that our subadditive convergence theorem will have further important applications for non-amenable entropy theory and beyond. In the framework of future reserach activities, it is planned to define topological cocycle entropy and investigate the validity of a variational principle in terms of relation invariant measures. Further projects might concern (cocycle) mean dimension and determining the entropy for algebraic actions.

Mathematics; Dynamical systems and ergodic theory; Global analysis, analysis on manifolds; Dynamical systems

video/mp4

Author affiliation: Techinon

2018-01-21

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64434

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/