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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 $A^\mathbb{Z}$.

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2 Answers 2

up vote 4 down vote accepted

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 for some flow-version examples. In particular see Proposition 1 and 2 there.

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Thanks...to clarify, is f a bump function? How does one choose f? –  Lewis Bowen Jul 7 '12 at 18:14
Yes, $f(x)$ is identically 1 outside a small neighborhood of $o$ and decreasing to 0 exponentially fast as $x\to o$. Then as $n\to \infty$, more and more mass of $\frac{1}{n}\sum\limits_{k=0}^{n-1}\delta_{\phi_kx}$ is concentrated at arbitrarily small neighborhood of $o$. I think it is not easy to give a simple characterization of conditions to ensure the transitivity of $\phi_1$. Intuitively, the homogeneity of flow induced by $X$ is broken: the orbit of $\phi_1$ is 'chaotic' and spread all over the manifold. –  Pengfei Jul 8 '12 at 2:27

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.

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