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 $A^\mathbb{Z}$.



Invariant measures and recurrent sets.
Just to clarify, the set of recurrent points may not be closed. There exists some transitive homeomorphism, whose uniquely ergodic measure is a Dirac measure. For example start with an irrational vector field $X$ on $\mathbb{T}^2$ and put a stop at $o\in\mathbb{T}^2$. That is, $Y=f\cdot X$ with $f(o)=0$. Then let $\phi\_1$ be the time-1 map of the flow induced by $Y$. We can choose $f$ such that $\delta\_o$ is the only invariant measure of $\phi\_1$, and $\phi\_1$ is transitive. In particular the set of recurrent points are dense on $\mathbb{T}^2$.


Edit: See the following [paper][1] for some flow-version examples. In particular see Proposition 1 and 2 there.


[1]: R. Saghin, W. Sun, E. Vargas. *On Dirac Physical Measures for Transitive Flows.* Commun. Math. Phys. 298, 741--756 (2010). <https://doi.org/10.1007/s00220-010-1077-9>


