The Rokhlin lemma for homeomorphisms of a Cantor set

For a Cantor set X, let Homeo(X) denote the group of all homeo-morphisms of X. The main result of this note is the following theorem. Several corollaries of this result are given. In particular, it is proved that for any aperiodic T ∈ Homeo(X) the set of all homeomorphisms conjugate to T is dense in the set of aperiodic homeomorphisms. 0. Introduction. One of the most useful results in ergodic theory which has many important applications is the Rokhlin lemma [R]. This statement asserts that given an aperiodic (non-singular) automorphism T of a standard measure space E are pairwise disjoint and µ(E ∪ T E ∪. .. ∪ T n−1 E) > 1 − ε. It immediately follows from this result that the set of periodic automorphisms is dense in the group of all nonsingular automorphisms of (X, B, µ) with respect to the metric d(T 1 , T 2) = µ{(x ∈ X : T 1 x = T 2 x)} where T 1 , T 2 ∈ Aut(X, B, µ). Subsequently, the Rokhlin lemma was generalized in various directions (see, for example, [AP, EP, FL, LW, OW]). It is well known that the Rokhlin lemma is also related to amenability. Our goal is to prove a version of the Rokhlin lemma in the context of Cantor dynamics. To start with, we need to consider a topology on Homeo(X) analogous to the metric d. It is well known that Homeo(X) equipped with the topology τ w