# invariant measures of homeomorphisms of the circle without periodic orbits

Here is a theorem of Furstenberg:

([2], Theorem 6.18 or [1], Theorem 1.3). If $T:\mathbb{R}/\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ is a homeomorphism without periodic orbits (i.e. $T^n(x)\neq x$ for any $n,x$) and $\mu$ is $T$-invariant Borel measure ($\mu(T^{-1}A)=\mu(A)$ for all Borel measurable $A$), then the function $F_{\mu}(x)=\mu([0,x])$ satisfies $F_{\mu}T(x)=T_{\alpha}F_{\mu}(x)\bmod 1$, where $T_{\alpha}x=x+\alpha\bmod 1$ is rotation by $\alpha=F_{\mu}T(0)$ (necessarily irrational).

My question is: which distribution functions can arise in this fashion? With $T$ as above, $\mu$ does not have any atoms (if $\mu(\{x\})>0$, then its orbit would have would have infinite measure by the assumption of no periodic orbits). Is this the only restriction? Does every continuous non-decreasing function $F:[0,1)\to[0,1)$ arise as $F_{\mu}$ for some $(T,\mu)$?

A related question: is every such $T$ (homeomorphism of the circle without periodic points) minimal, i.e. is every orbit dense? If so, then $F_{\mu}$ above is a homeomorphism. I assume this isn't the case otherwise the statement of [2], Theorem 6.18 would be different).

I'd also be happy with an example of a $T$ such that $F_{\mu}$ is not a homeomorphism, probably supported on something Cantor-like if it exists (which I think it does).

Thanks!

[1] H. Furstenberg, $\textit{Strict ergodicty and transformations of the torus}$, Amer. J. Math. 83, 1961, $573-601$

[2] Peter Walters, $\textit{An Introduction to Ergodic Theory}$, Springer

• So F is saying that $T$ is conjugate to an irrational shift mod $1$, where $x\to F_\mu(x)$ is the conjugating map? – kimchi lover Jul 20 '17 at 1:28
• @kimchilover yes (I would've drawn a commutative diagram, but I didn't know how). – yoyo Jul 20 '17 at 21:34

For starters, it looks like any atomless $\mu$ for which $x\to\mu([0,x])$ is a homeomorphism will do. I think F's result does not cover the case where $\mu$ is the Cantor measure, though: even though $F_\mu: K\to \mathbb R /\mathbb Z$ is continuous and surjective as a map on the Cantor set, pushing $\mu$ to Lebesgue measure, I don't see how to extend it to a homeomorphism $\mathbb R /\mathbb Z\to \mathbb R /\mathbb Z.$