1

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.

Rick Sanchez C-666
  • 1,010
  • The statement is true if we identify the endpoints of the interval to get the unit circle; see https://math.stackexchange.com/q/4383186/169085. – Alp Uzman Feb 16 '22 at 04:24

1 Answers1

2

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 $n \to \infty$.

Adam
  • 2,677