Suppose $T:X \to X$ is a homeomorphism of a compact metric space. The recurrent set is the set of all points $x \in X$ such that for every $\epsilon>0$ there exists an $n\in \mathbb{Z}$, $n \ne 0$, such that $d(T^nx,x)<\epsilon$. Does there exists a $T$-invariant probability measure $\mu$ on $X$ with support equal to the recurrent set?

A few notes: 1) By the Poincare recurrence theorem, the support of any invariant measure is always contained in the recurrent set but it might be properly contained.

2) It's easy to prove that this is true for symbolic systems. That is, if $A$ is a finite set, $T$ is the shift map on $A^\mathbb{Z}$ and $X$ is a closed shift-invariant subset of $A^\mathbb{Z}$.