Construction of an analogue of Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings. (English. Russian original) Zbl 0821.58027

Volevich, V. A.

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

Keywords:

spatio-temporal chaos; symbolic dynamics; coupled map lattice; Sinai- Ruelle-Bowen measure; Gibbs random fields