Title: Finite generating partitions for continuous actions of countable groups
Speaker: Anush Tserunyan (UIUC)
Abstract: Let a countable group G act continuously on a Polish space X. A countable Borel partition P of X is called a generator if the set of its translates {gA : g in G, A in P} generates the Borel sigma-algebra of X. For G=Z, the Kolmogorov-Sinai theorem gives a measure-theoretic obstruction to the existence of finite generators: they don't exist in the presence of an invariant probability measure with infinite entropy. It was asked by B. Weiss in the late 80s whether the nonexistence of any invariant probability measure guarantees the existence of a finite generator. We show that the answer is positive for an arbitrary countable group G and sigma-compact X (in particular, for locally compact X). We also show that finite generators always exist for aperiodic actions in the context of Baire category (i.e. allowing ourselves to disregard a meager set), thus answering a question of A. Kechris from the mid-90s.
