# Measure-theoretic equicontinuity and rigidity

## TL;DR

This paper characterizes measure-theoretic rigidity in topological dynamical systems through the lens of measure-$A$-equicontinuity and mean-equicontinuity, linking these properties to subsequences and IP-sets.

## Contribution

It establishes new equivalences between rigidity and measure-$A$-equicontinuity, including for IP-sets and positive density subsequences, and extends results to functions.

## Key findings

- Rigidity is equivalent to existence of certain measure-$A$-equicontinuity subsequences.
- Positive density measure-$A$-mean-equicontinuity implies rigidity.
- Results extend to functions within the dynamical systems context.

## 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.