MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this community
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a single point then there's only one such $\mu$. Also, if $f$ is constant.
So assume $X$ is the Cantor space $\{0,1\}^{\mathbb N}$, or the unit interval $[0,1]$, and $f$ is invertible.
Must there be uncountably many such $\mu$? Must there be an uncountable family of mutually singular $\mu$'s?
Not at all. Take $f(x)=x^2$ on the unit interval. For examples of uniquely ergodic homeomorphisms of the Cantor set see The prevalence of uniquely ergodic systems by Jewett. A simple explicit example is provided by the boundary action of any hyperbolic automorphism of a homogeneous tree (attach a binary tree to each integer and look at the action of the integer shift on the space of ends of the whole construction).
