Measure-theoretic equicontinuity and rigidity

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

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