The Sinai-Bowen-Ruelle measure for a multidimensional lattice of interacting hyperbolic mappings.

*(English. Russian original)*Zbl 0823.58025
Russ. Acad. Sci., Dokl., Math. 47, No. 1, 117-121 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 328, No. 5, 540-543 (1993).

Let \(k\geq 1\) be an integer and \(M\) a compact manifold. For the space \(M^ \infty= \prod_{k\in \mathbb{Z}^ d} M_ k\) (all \(M_ k\) diffeomorphic to \(M\)) topologized by the maximum metric mappings \(\phi: M^ \infty\to M^ \infty\) are considered which can be obtained from the combination of individual mappings \(f_ k: M_ k\to M_ k\) \((k\in \mathbb{Z}^ d)\) each of which is Anosov by adding a small interaction \(A\) (i.e., \(\phi(\{z_ k\})= A(\{f_ k(x_ k)\})\) where \(A: M^ \infty\to M^ \infty\) is close to the identity, and if \(x= \{x_ k\}\), \(y= \{y_ k\}\in M^ \infty\) are such that \(x_ k= y_ k\) with only one exception at \(k= k_ 0\), then the distance between \(A(x)_{k'}\) and \(A(y)_{k'}\) in \(M_{k'}\) becomes exponentially small if \(| k'- k_ 0|\) grows to infinity, and a similar condition holds for the first and second derivatives).

Then the main result states the existence of a probability measure \(\mu\) on \(M^ \infty\) which has the following properties of an SBR measure:

(i) \(\mu\) is invariant under \(\phi\) and under the shifts \(\{x_ k\}\to \{x_{k- k_ 0}\}\) for \(k_ 0\in \mathbb{Z}^ d\) fixed.

(ii) \(\phi\) and the shifts are mixing on \(M^ \infty\).

(iii) For certain measures \(\nu\) on \(M^ \infty\) with exponential decay of spatial correlations (which replace the absolutely continuous measures in the definition of SBR measures in the finite-dimensional case) \(\phi^{* m}\nu\) converges weakly to \(\mu\) for \(m\to\infty\).

Then the main result states the existence of a probability measure \(\mu\) on \(M^ \infty\) which has the following properties of an SBR measure:

(i) \(\mu\) is invariant under \(\phi\) and under the shifts \(\{x_ k\}\to \{x_{k- k_ 0}\}\) for \(k_ 0\in \mathbb{Z}^ d\) fixed.

(ii) \(\phi\) and the shifts are mixing on \(M^ \infty\).

(iii) For certain measures \(\nu\) on \(M^ \infty\) with exponential decay of spatial correlations (which replace the absolutely continuous measures in the definition of SBR measures in the finite-dimensional case) \(\phi^{* m}\nu\) converges weakly to \(\mu\) for \(m\to\infty\).

Reviewer: H.G.Bothe (Berlin)