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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?

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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).

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  • $\begingroup$ Impressive speed! I suppose something similar must work for the Cantor space? (I suppose that could be a separate question) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 27, 2018 at 22:37
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    $\begingroup$ I have added an edit about the Cantor set case. $\endgroup$
    – R W
    Commented Mar 27, 2018 at 22:59
  • $\begingroup$ That's great. Another link is jstor.org/stable/24901717?seq=1#page_scan_tab_contents $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Mar 27, 2018 at 23:08

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