Finite generators for countable group actions in the Borel and Baire category settings

Details Finite generators for countable group actions in the Borel and Baire category settings Finite generators for countable group actions in the Borel and Baire category settings

2012-04-03

arxiv.org/abs/1204.0829v1

Mathematics

Logic

Scientific paper

For a countable group G and a standard Borel G-space X, a countable Borel partition P of X is called a generator if {gA : g in G, A in P} generates the Borel sigma-algebra of X. For G=Z, the Kolmogorov-Sinai theorem implies that if X admits an invariant probability measure with infinite entropy, then there is no finite generator. It was asked by Benjamin Weiss in '87 whether removing this obstruction would guarantee the existence of a finite generator; more precisely, if X does not admit an invariant probability measure at all, is there a finite generator? We give a positive answer to this question for arbitrary G in case X has a sigma-compact topological realization (e.g. when X is a sigma-compact Polish G-space). Assuming a positive answer to Weiss's question for arbitrary Borel Z-spaces, we prove two dichotomies, one of which states the following: for an aperiodic Borel Z-space X, either X admits an invariant probability measure of infinite entropy or there is a finite generator. We also show that finite generators always exist in the context of Baire category, thus answering positively a question raised by Kechris in the mid-'90s. More precisely, we prove that any aperiodic Polish G-space admits a 4-generator on an invariant comeager set. It is not hard to prove that finite generators exist in the presence of a weakly wandering or even just a locally weakly wandering complete section. However, we develop a sufficient condition for the nonexistence of non-meager weakly wandering subsets of a Polish Z-space, using which we show that the nonexistence of an invariant probability measure does not guarantee the existence of a countably generated Baire measurable partition into invariant sets that admit weakly wandering complete sections. This answers negatively a question posed by Eigen, Hajian and Nadkarni, which was also independently answered by Ben Miller.

Affiliated with

Also associated with

No associations

Rating

Finite generators for countable group actions in the Borel and Baire category settings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Finite generators for countable group actions in the Borel and Baire category settings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Finite generators for countable group actions in the Borel and Baire category settings will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-31223