Generically there is but one self homeomorphism of the Cantor set
Ethan Akin, Eli Glasner, Benjamin Weiss
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 Gδ subset of H(X). We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.
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