Volevich, V. A. 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) Cited in 2 Documents 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 Keywords:spatio-temporal chaos; symbolic dynamics; coupled map lattice; Sinai- Ruelle-Bowen measure; Gibbs random fields PDF BibTeX XML Cite \textit{V. A. Volevich}, Russ. Acad. Sci., Sb., Math. 79, No. 2, 1 (1993; Zbl 0821.58027); translation from Mat. Sb. 184, No. 8, 17--36 (1993) Full Text: DOI