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Construction of an analogue of Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings. (English. Russian original) Zbl 0821.58027
Russ. Acad. Sci., Sb., Math. 79, No. 2, 347-363 (1994); translation from Mat. Sb. 184, No. 8, 17-36 (1993).
The author considers a coupled map lattice, that is a map of $$M^{\mathbb{Z}^ d}$$ where $$M$$ is a compact Riemannian manifold, which is a perturbation of the pointwise product of a single Anosov map $$f$$ of $$M$$. Specifically, let $$F: M^{\mathbb{Z}^ d} \rightarrow M^{\mathbb{Z}^ d}$$ be defined by $$F(x)_ k = f(x_ k)$$ for $$k \in \mathbb{Z}^ d$$ and $$x \in M^{\mathbb{Z}^ d}$$. Let $$G : M^{\mathbb{Z}^ d} \rightarrow M^{\mathbb{Z}^ d}$$ be a ‘weak coupling’ (that is a translation-commuting $$C^ 2$$ map which is close in the $$C^ 2$$ topology to the identity). Then the composition $$G \circ F$$ is a coupled map lattice. In the case where $$G$$ is the identity, the mapping behaves as infinitely many independent copies of the original dynamical system $$(M,f)$$, which is known to possess a Sinai-Ruelle-Bowen measure. The system $$(M^{\mathbb{Z}^ d},F)$$ then possesses a Sinai-Ruelle-Bowen measure which additionally is translation-invariant. The author shows that for $$G$$ sufficiently close to the identity, the existence of a translation-invariant Sinai-Ruelle- Bowen measure persists.
The proof involves the construction of local stable and unstable manifolds and construction of the measure on these. The author then introduces a symbolic dynamical representation and uses this to make use of the techniques from the theory of Gibbs random fields to complete the proofs.
Reviewer: A.Quas (Cambridge)

##### MSC:
 37A99 Ergodic theory 37D99 Dynamical systems with hyperbolic behavior 28D05 Measure-preserving transformations 60G60 Random fields 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 37C75 Stability theory for smooth dynamical systems
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