Measure-theoretic equicontinuity and rigidity

Abstract: Let $(X,T)$ be a topological dynamical system and $\mu$ be a invariant measure, we show that $(X,\mathcal{B},\mu,T)$ is rigid if and only if there exists some subsequence $A$ of $\mathbb N$ such that $(X,T)$ is $\mu$-$A$-equicontinuous if and only if there exists some IP-set $A$ such that $(X,T)$ is $\mu$-$A$-equicontinuous. We show that if there exists a subsequence $A$ of $\mathbb N$ with positive upper density such that $(X,T)$ is $\mu$-$A$-mean-equicontinuous, then $(X,\mathcal{B},\mu,T)$ is rigid. We also give results with respect to functions.