Transfer operators for coupled map lattices

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Abstract
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 τ: [0, 1] → [0, 1], which is the same on each component. Denote the direct product of these identical systems by T: X → X where X = [0, 1]L. From this product system we obtain a coupled map lattice (CML) Sε: X → 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 ε.For a broad class of piecewise expanding single-component-transformations τ 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.

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