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The rotation $Y$ is an almost $1$-to-$1$ factor of the Sturmian $X$ in the sense of Hochman [1], meaning for any invariant probability measure of $Y$ the set of points with unique preimage has full me …

This is just a comment on your definition of "uniformly recurrent". Can you give an example of a system you'd consider uniformly recurrent? If this notion is standard, the following must be nonsense, …

I am doubtful about your claim 2): I think there do exist subshifts where some recurrent points are not in the support of any probability measure. Let $X$ be the subshift associated to the substitutio …

Here's a subshift counterexample. Let $E : \{0,1\}^* \to \{0,1\}^*$ be the map on finite words that flips every second bit, preserving word length, e.g. $E(01000) = 11101$, and let $O$ flip the even p …

The answer is no, I think any of the usual examples works. Some argument below. Lemma. Suppose $X$ is a compact metric space, $\mu$ a nonatomic probability measure on $X$, and $T : X \to X$ is a mini …

edit As the author clarifies in the comments below, there is indeed a hidden assumption. In the case of a subshift, as Vaughn Climenhaga explains the extra condition means exactly that the subshift is …

This is joint work with Ilkka Törmä (50-50, I am just more interested in points than him). We did not see a link to cohomology, but your technical question was very interesting, and we believe we solv …

It's indeed unbounded for every irrational $x$. Let me identify points of $\mathbb{R}/\mathbb{Z}$ with their representatives on $[0,1)$, and order it by the usual order $<$ of $\mathbb{R}$ applied to …

So, I probably did not initially understand you correctly. Let me analyze four interpretations of your construction; the first is what I thought first, the second gives something uninteresting, the th …