The group of homeomorphisms of the Cantor set has ample generics

Details The group of homeomorphisms of the Cantor set has ample generics The group of homeomorphisms of the Cantor set has ample generics

2011-04-17

arxiv.org/abs/1104.3340v2

Mathematics

Dynamical Systems

final version, to appear in Bulletin of the London Mathematical Society

Scientific paper

We show that the group of homeomorphisms of the Cantor set $H(K)$ has ample generics, that is, for every $m$ the diagonal conjugacy action $g\cdot(h_1,h_2,..., h_m)=(gh_1g^{-1},gh_2g^{-1},..., gh_mg^{-1})$ of $H(K)$ on $H(K)^m$ has a comeager orbit. This answers a question of Kechris and Rosendal. We show that the generic tuple in $H(K)^m$ can be taken to be the limit of a certain projective Fraisse family. We also present a proof of the existence of the generic homeomorphism of the Cantor set in the context of the projective Fraisse theory.

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