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Transfer operators for coupled map lattices. (English) Zbl 0737.58032
Let \(L\) denote a finite or infinite one-dimensional lattice. To each lattice site is attached a copy of a dynamical system with phase space \([0,1]\) and dynamics described by a transformation \(\tau: [0,1]\to [0,1]\), which is the same on each component. Denote the direct product of these identical systems by \(T: X\to X\) where \(X=[0,1]^ L\). From this product system we obtain a coupled map lattice (CML) \(S\epsilon: X\to X\), if we introduce some interaction between the components, e.g. by averaging between nearest neighbours. The strength of the coupling depends upon some parameter \(\epsilon\).
For a broad class of piecewise expanding single-component-transformations \(\tau\) we study such systems via their transfer operators and treat the coupled system as a perturbation of the uncoupled one. This yields existence and stability results for \(T\)-invariant measures with absolutely continuous finite-dimensional marginals. For a more restricted class of CML’s, L. A. Bunimovich and Ya. G. Sinaj [Nonlinearity 1, No. 4, 491-516 (1988; Zbl 0679.58028)] obtained existence and uniqueness results using methods from statistical mechanics.

MSC:
37A99 Ergodic theory
28D05 Measure-preserving transformations
37E99 Low-dimensional dynamical systems
37D99 Dynamical systems with hyperbolic behavior
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