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 $\mu$. But if $X$ is say the Cantor space $\{0,1\}^{\mathbb N}$, or the unit interval $[0,1]$,

must there be uncountably many such $\mu$? Must there be an uncountable family of mutually singular $\mu$'s?

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?