Ergodic theory question about the support of a measure.

Question

I am trying to work through some problems from " Ergodic theory with view towards number theory" but I am finding it a tad more difficult than expected. In particular, I have the following question:

For $X$ a compact metric space, $T:X\to X$ continuous and $\mu$ a $T$-invariant ergodic probability measure defined on the Borel sets of $X$. Prove that for $\mu$ almost every $x\in X$ and $y\in \operatorname{Supp}(\mu)$ there exists a sequence $n_k\nearrow\infty$ such that $T^{n_k}(x)\to y$. Recall $$\operatorname{Supp}(\mu)=X-\bigcup U$$ where $U\subset X$ is open and has null measure.

This problem screams the Poincaré Recurrence Theorem to me, but I am not sure where to use it. I could really use some guidance for this problem.

NOTE: This is not homework for a class, I just really want to learn ergodic theory.

Answer 1

1. It is enough to prove that for each $y\in\operatorname{Supp}\mu$, we have $$\mu\{x\in X\mid\exists n_k\uparrow +\infty, T^{n_k}x\to y\}=1.$$ Indeed, assume we have done that. The support of $\mu$ is a closed set in a separable metric space, hence it is separable. Let $\{y_j,j\in\Bbb N\}$ be a countable dense subset. As a countable union of negligible sets is negligible, we have $$\mu\left\{x\in X\mid\forall j\in\Bbb N,\exists n_k\uparrow +\infty, T^{n_k}x\to y_j\right\}=1.$$ Let $y\in \operatorname{Supp}(\mu)$. Then for each $i$, consider $y_{j_i}$ for which $d(y,y_{j_i})<i^{-1}$, and construct $(n_i)$ by induction, taking $d(y_{j_i},T^{n_i}x)<i^{-1}$.

2. Fix now $y\in\operatorname{Supp}\mu$, and call $C:=\left\{x\in X\mid\exists n_k\uparrow +\infty, T^{n_k}x\to y\right\}$. We can check that $C$ is almost invariant, in sense that $\mu(C\Delta T^{-1}C)=0$. As $T$ is ergodic, the measure of $C$ is either $0$ or $1$.

3. As $y$ is in the support of $\mu$, for each $l$, $\mu\left(B\left(y,l^{-1}\right)\right)$ is positive. By Poincaré recurrence theorem, $\mu\left( \left\{x\in B\left(y,l^{-1}\right) \mid T^nx\in B\left(y,l^{-1}\right)\mbox{ for infinitely many }n\mbox{'s} \right\} \right)$ is positive hence, by ergodicity, its value is $1$. Therefore, the measure of $C$ is $1$.