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Rohlin properties for $ \mathbb{Z}^{d}$ actions on the Cantor set

Author: Michael Hochman
Journal: Trans. Amer. Math. Soc. 364 (2012), 1127-1143
MSC (2010): Primary 37C50, 37C85, 37B50, 54H20; Secondary 03D99
Published electronically: October 18, 2011
MathSciNet review: 2869170
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Abstract: We study the space $ \mathcal {H}(d)$ of continuous $ \mathbb{Z}^{d}$-actions on the Cantor set, particularly questions related to density of isomorphism classes. For $ d=1$, Kechris and Rosendal showed that there is a residual conjugacy class. We show, in contrast, that for $ d\geq 2$ every conjugacy class in $ \mathcal {H}(d)$ is meager, and that while there are actions with dense conjugacy class and the effective actions are dense, no effective action has dense conjugacy class. Thus, the action by the group homeomorphisms on the space of actions is topologically transitive but one cannot construct a transitive point. Finally, we show that in the spaces of transitive and minimal actions the effective actions are nowhere dense, and in particular there are minimal actions that are not approximable by minimal shifts of finite type.

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Additional Information

Michael Hochman
Affiliation: Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey 08544
Address at time of publication: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University, Jerusalem 91904, Israel

Received by editor(s): January 16, 2009
Received by editor(s) in revised form: January 12, 2010
Published electronically: October 18, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.