Measure-theoretic equicontinuity and rigidity of group actions

Abstract

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 $f\in L^2(\mu )$ with respect to an infinite sequence Ω of G. Then we give some equivalent conditions for $f\in L^2(\mu )$ 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.

Keywords:

• Measure-theoretic equicontinuity
• rigidity
• measure-theoretic mean equicontinuity

2020 Mathematics Subject Classifications:

• 37A05
• 37B02

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 12061043 and 11661054].