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Generically there is but one self homeomorphism of the Cantor set

Author(s): Ethan Akin; Eli Glasner; Benjamin Weiss
Journal: Trans. Amer. Math. Soc. 360 (2008), 3613-3630.
MSC (2000): Primary 22A05, 22D05; Secondary 54C40, 37E15
Posted: February 27, 2008
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Abstract: We describe a self homeomorphism $ R$ of the Cantor set $ X$ and then show that its conjugacy class in the Polish group $ H(X)$ of all homeomorphisms of $ X$ forms a dense $ G_\delta$ subset of $ H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.

References:

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E. Akin, Good measures on Cantor space, Trans. Amer. Math. Soc. 357, (2005), 2681-2722. MR 2139523 (2006e:37003)

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E. Akin, M. Hurley and J. Kennedy, Dynamics of topologically generic homeomorphisms, Mem. Amer. Math. Soc. 164, (2003), no. 783. MR 1980335 (2004j:37024)

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E. Glasner and B. Weiss, The topological Rohlin property and topological entropy, Amer. J. Math. 123, (2001), 1055-1070. MR 1867311 (2002h:37025)

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A. S. Kechris and C. Rosendal, Turbulence, amalgamation and generic automorphisms of homogeneous structures, Proc. London Math. Soc. 94, (2007), 302-350. MR 2308230

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D.V. Osin, Small cancellations over hyperbolic groups and embedding theorems, arXiv:math.GR/0411039v1, 2 Nov 2004.

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J. Oxtoby, Measure and category ($ 2^{nd}$ Ed.) Springer-Verlag, Berlin, 1980. MR 584443 (81j:28003)

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Additional Information:

Ethan Akin
Affiliation: Mathematics Department, The City College, 137 Street and Convent Avenue, New York, New York 10031
Email: ethanakin@earthlink.net

Eli Glasner
Affiliation: Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
Email: glasner@math.tau.ac.il

Benjamin Weiss
Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israel
Email: weiss@math.huji.ac.il

DOI: 10.1090/S0002-9947-08-04450-4
PII: S 0002-9947(08)04450-4
Keywords: Rohlin property, group of homeomorphisms of the Cantor set, conjugacy class
Received by editor(s): April 26, 2006
Posted: February 27, 2008
Additional Notes: This research was supported by ISF grant # 1333/04.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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