TRANSFER OPERATORS FOR COUPLED MAP LATTICES

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] --> [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) S(epsilon):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-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.