We investigate the genericity of measure-preserving actions of the free group Fn,
on possibly countably infinitely many generators, acting on a standard probability
space. Specifically, we endow the space of all measure-preserving actions of Fn acting
on a standard probability space with the weak topology and explore what properties
may be verified on a comeager set in this topology. In this setting we show an analog
of the classical Rokhlin Lemma. From this result we conclude that every action of Fn
may be approximated by actions which factor through a finite group. Using this finite
approximation we show the actions of Fn, which are rigid and hence fail to be mixing,
are generic. Combined with a recent result of Kerr and Li, we obtain that a generic
action of Fn is weak mixing but not mixing. We also show a generic action of Fn has
sigma-entropy at most zero. With some additional work, we show the finite approximation
result may be used to that show for any action of Fn, the crossed product embeds
into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing
the finite approximation result may be transferred to a subspace of the space of all
topological actions of Fn on the Cantor set. Within this class, we show the set of
actions with sigma-entropy at most zero is generic.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-2010-08-8418 |
| Date | 2010 August 1900 |
| Creators | Hitchcock, James Mitchell |
| Contributors | Kerr, David |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | thesis, text |
| Format | application/pdf |
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