Invariant measures and recurrent sets.

Question

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 $A^\mathbb{Z}$.

Answer 1

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

Answer 2

I am doubtful about your claim 2): I think there do exist subshifts where some recurrent points are not in the support of any probability measure.

Let $X$ be the subshift associated to the substitution $0 \mapsto 000$, $1 \mapsto 101$, i.e. the $\mathbb{Z}$-subshift where $x \in X \iff \forall u \sqsubset x: \exists i: u \sqsubset \tau^i(1)$, where $\sqsubset$ means "appears as subword" (subwords meaning consecutive positions here).

A short calculation (e.g. eigenvalue decomposition) shows that the density of $1$s in $\tau^i(1)$ tends to $0$. Since all subwords of $\tau^j(1)$ with $j \geq i$ can be decomposed into $\tau^i(1)$s separated by $0$s, we see that the upper density of $1$s is $0$ in every point of the subshift.

Let $y$ denote the one-sided limit $\lim_i \tau^i(1) \in \{0,1\}^{\mathbb{N}}$. It is easy to see that $x = ....000.y$ (an infinite left tail of $0$s followed by right tail $y$) is in $X$, and is a recurrent point.

This point $x$ is not in the support of any measure: suppose it is in the support of $\mu$. Then $\mu([1]) > 0$, and by ergodic decomposition where is an ergodic measure $\nu$ with $\nu([1]) > 0$. Generic points for such $\nu$ cannot exist since the density of $1$s in every point in $X$ is $0$.

Answer 3

In the class of Cantor dynamical systems, one can easily construct examples of aperiodic homeomorphisms $T : X \to X$ with one minimal component, $Y$, such that the following conditions hold: (i) $T$ is (minimal) uniquely ergodic on $Y$; (ii) $T$ acts minimally on $X\setminus Y$ (every orbit is dense) and has a unique probability invariant measure $\mu$ sitting on $X \setminus Y$. So that $T$ has exactly two ergodic invariant measures.

The above example can be realized, for example, for an aperiodic substitution dynamical system.