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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 $$m\to\infty$$.
Reviewer: H.G.Bothe (Berlin)

##### MSC:
 37A99 Ergodic theory 37D99 Dynamical systems with hyperbolic behavior 28D05 Measure-preserving transformations 60B10 Convergence of probability measures 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics