Timeline for Proof that Sturmian shift is uniquely ergodic using irrational rotation
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 8, 2022 at 5:58 | answer | added | Ville Salo | timeline score: 1 | |
| Jun 7, 2022 at 20:56 | comment | added | Ville Salo | 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. | |
| Jun 7, 2022 at 20:55 | comment | added | Ville Salo | 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 $0$. | |
| Jun 7, 2022 at 15:38 | comment | added | Christian Remling | 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. | |
| Jun 7, 2022 at 3:01 | history | asked | kiki | CC BY-SA 4.0 |