On the finite-dimensional marginals of shift-invariant measures
| Autor: | Chazottes, J. -R., Gambaudo, J. -M., Hochman, M., Ugalde, E. |
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| Rok vydání: | 2010 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $\Sigma$ be a finite alphabet, $\Omega=\Sigma^{\mathbb{Z}^{d}}$ equipped with the shift action, and $\mathcal{I}$ the simplex of shift-invariant measures on $\Omega$. We study the relation between the restriction $\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes $\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant" measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of the convex set $\mathcal{I}_n$ which turns out to be strikingly different when $d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also extreme points which arise in other ways. We show that $\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they differ for $n$ large enough. We also show that while in dimension one $\mathcal{I}_n$ are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of $\mathcal{I}_n$ for all large enough $n$. Comment: 20 pages, 3 figures, to appear in Ergod. Th. & Dynam. Sys |
| Databáze: | arXiv |
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