I am finding proofs of unique ergodicity of Sturmian shifts however I want to know if there is a proof that link that to the unique ergodicity of irrational rotations through conjugacy for example or by another way ?
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$\begingroup$ Sturmian (or other) shifts are systems on subspaces $X\subseteq \{ 0,1\}^{\mathbb Z}$, while rotations act on $Y=S^1$. Since $X,Y$ are not homeomorphic, there is no conjugacy. $\endgroup$– Christian RemlingCommented Jun 7, 2022 at 15:38
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$\begingroup$ The rotation $Y$ is an almost 1-to-1 factor of the Sturmian $X$, in the sense that for any shift-invariant probability measure of $Y$ the set of points with unique preimage has full measure. Namely, there is a single rotation orbit whose elements have two preimages, and any shift-invariant probability measure will clearly give it measure $0$. $\endgroup$– Ville SaloCommented Jun 7, 2022 at 20:55
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$\begingroup$ In this situation, there is a bijection between shift-invariant Borel probability measures on $X$ and $Y$. See [Hochman, Michael. "On the dynamics and recursive properties of multidimensional symbolic systems." Inventiones mathematicae 176.1 (2009): 131-167]. So indeed from the unique ergodicity of $Y$ you can conclude the same for $X$ using not quite conjugacy but the fact the factor map forgets very little. $\endgroup$– Ville SaloCommented Jun 7, 2022 at 20:56
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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 measure. Namely, there is a single rotation orbit whose elements have two preimages (basically by definition of the coding), so any invariant probability measure will clearly give it measure $0$.
Under the almost $1$-to-$1$ assumption, there is a bijection between shift-invariant Borel probability measures on $X$ and $Y$ [1] (I don't know in which situations exactly, but at least it's true in the compact metric setting). Proof: Let $Z \subset Y$ be the points with multiple preimages. Note that it's Borel and invariant, and by assumption invariant measures on $Y$ give it measure $0$. If you have an invariant measure on $X$, you can push one forward to $Y$ directly; if you have one on $Y$, because the measure of $Z$ is $0$ you can pull it back in an equally natural way, and these are inverses. I imagine this gives even an affine homeomorphism between the corresponding Choquet simplices, though I haven't thought too deeply about the subtleties.
So indeed from the unique ergodicity of $Y$ you can conclude the same for $X$ using not quite conjugacy but the fact the factor map forgets very little.
Note that to have the measure correspondence it's important to use Hochman's definition. In the topological category one usually requires that the set of points with multiple preimages be topologically small. Calling this a topologically almost $1$-to-$1$ extension, the main result of [2] states that every extension of a system $Y$ is, from the measure-theoretic point of view, the same as some topologically almost $1$-to-$1$ extension.
[1] Hochman, Michael, On the dynamics and recursive properties of multidimensional symbolic systems, Invent. Math. 176, No. 1, 131-167 (2009). ZBL1168.37002.
[2] Furstenberg, Hillel; Weiss, Benjamin, On almost 1-1 extensions, Isr. J. Math. 65, No. 3, 311-322 (1989). ZBL0676.28010.
