I'm trying to solve the following problem:
Let $T\colon X\to X$ be a continuous map on a compact metric space $X$, uniquely ergodic. Let $Y\neq \emptyset$ be an open set. Show that $t(x) = \min\left\{n\ge 1 : T^n x\in Y\right\}$ is well defined and bounded.
I was able to prove it assuming that the measure has full support:
Thanks to unique ergodicity, for every continous map $f$, $\frac 1 n \sum_{k=0}^{n-1} f(T^k x) \to c=\int_X f$ uniformly.
If $t$ isn't bounded, then for each $n$ I can find a $x_n$ such that $\frac 1 n \sum_{k\le n-1} f(T^k x) = 0$, and $c=0$. In particular if I take $f\ge0$ to be 1 on an open set inside $Y$ and $0$ outside of $Y$, then I have a contradiction assuming that the measure has full support!
Do I need to assume that? Or is there another way? Taking such a $f$ was a hint given in the exercise, but for example if $\mu(Y) =0$ I don't know what to do. In fact I don't know any example where there is unique ergodicity and where some open sets have 0 measure, I'm also interested in such examples.
Thanks for any help
Best Answer
Here's a counter-example for what you are trying to prove. As your question hints at, this is a very simple example where unique ergodicity holds yet some nonempty open set has measure zero.
Take $X = \mathbb R \cup \{P\}$ to be the one-point compactification of the real line $\mathbb R$. Define $f : X \to X$ by the formula $$f(x) = \begin{cases} P & \quad\text{if $x=P$} \\ x+1 & \quad\text{if $x \in \mathbb R$} \end{cases} $$ The only invariant probability measure for this example is the Dirac measure on the point $P$, so $f$ is uniquely ergodic, but $\mathbb R = X - \{P\}$ is a nonempty invariant open subset.
Now define $Y = (-1,\infty) \subset \mathbb R$, which is an open subset of $X$. Note that $t(P) = +\infty$, and perhaps this might be what you would mean by saying that $t(P)$ is "not well-defined". And note also that while $t(x)$ is indeed well-defined for $x \in \mathbb R$, and while $t(x)=0$ for $x > -1$, on the other hand for any natural number $n$ any value of $x \in (-\infty,-n]$ satisfies $t(x) \ge n$, and so $t(x)$ has no upper bound.
My guess as to what is going on here is that you need to simply add some kind of topological hypothesis to your problem, for example that $f$ is topologically minimal, meaning that there is no proper invariant closed subset.