Let $f:[0,1]\to[0,1]$ be the continuous map $$f(x)=x+\cfrac{1}{4}\sin^2(\pi x)$$ Is $f$ uniquely ergodic?
Note: If $X$ is a compact metric space and $f : X \rightarrow X$ a continuous map, then $f$ is said to be uniquely ergodic if
$$\cfrac{1}{n}\sum_{k=0}^{n-1}\phi\circ f^k$$
converges to a constant uniformly for every continuous function $\phi$.
I suppose one way to approach this problem is to show that the time averages converge to the value at $0$, but I'm not sure how to make it work here.
Any hints? Thank you.
Best Answer
Note $f(0) = 0$ and $f(1) = 1$. So if you take $\phi$ to be the identity then $n^{-1} \sum_{k=0}^{n-1} \phi(f^k(0)) = 0$ and $n^{-1} \sum_{k=0}^{n-1} \phi(f^k(1)) = 1$ for all $n$. Therefore this expression is not converging uniformly to a constant as $n \to \infty$.