Logic Seminar
Anush Tserunnyan
UCLA
Finite generators for countable group actions
Abstract: Consider a Borel action of a countable group G on a standard Borel space X. A countable Borel partition P of X is
called a generator if GP={gA: g in G, A in P} generates the Borel sigma-algebra of X.
Existence of such P of cardinality n is equivalent to the existence of a G-embedding of X into the shift n^G. For G=Z, the Kolmogorov-Sinai theorem
implies that finite generators don't exist in the presence of an invariant probability measure with infinite entropy.
It was asked by Weiss in the late 80s, whether the nonexistence of any invariant probability measure would guarantee the existence of a finite generator.
We show that the answer is positive in case X admits a sigma-compact topological realization (e.g. if X is a sigma-compact Polish G-space).
We also show that finite generators always exist in the context of Baire category thus answering a question
of Kechris. In fact, we show that if X is a Polish G-space having infinite orbits, then there is a
4-generator on an invariant comeager set.
Wednesday April 25, 2012 at 3:00 PM in SEO 427