Measure-theoretic equicontinuity and rigidity of group actions

Let (G,X) be a G-system, which means that X is a compact Hausdorff space and G is an infinite topological group continuously acting on X, and let μ be a G-invariant measure of (G,X). In this paper, we introduce the concepts of rigidity, uniform rigidity and μ-Ω-equicontinuity of (G,X) with respect to an infinite sequence Ω of G and the notions of μ-Ω-equicontinuity and μ-Ω-mean-equicontinuity of a function f∈L2(μ) with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for f∈L2(μ) and (G,X) to be rigid, respectively. In addition, if G is commutative and X satisfies the first axiom of countability, we present some equivalent conditions for (G,X) to be uniformly rigid.