Let $(X, \mathcal{X}, \mu, T)$ be a measure preserving system. In this setting, $X$ is a (compact if need be) topological space, $\mathcal{X}$ is a Borel sigma algebra, and $T$ continuous. Let $A \in \mathcal{X}$ be a measurable set with positive measure $\mu(A) > 0$. Let $T_A : A \to A$ be the first return map.
Now, we say that $T$ equidistributes if for all compactly supported continuous functions $f \in C_c(X)$ and for $\mu$-a.e. $x \in X$ we have that $$\frac{1}{N} \sum_{k=0}^{N-1} f(T^nx) \to \int_X f\, d\mu.$$
Is there a relationship between $T$ and $T_A$ when it comes to equidistribution?