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110
T. Fischer and H. H. Rugh
variation and prove the existence of an invariant probability measure. For the infinite-
dimensional system they further show that for a small perturbation of the uncoupled map
any invariant measure in BV is close (in a specified sense) to what they found. Coupled
map lattices with multi-dimensional local systems of the hyperbolic type have been studied
by Pesin and Sinai [16], Jiang [8, 9], Jiang and Mazel [10], Jiang and Pesin [11] and
Volevich [18, 19]. Detailed surveys on coupled map lattices can be found in [5, 11, 3].
In the above papers (except [1, 13]) the analysis has been performed only for Banach
spaces defined for finite subsets 3 of the lattice, and the (weak) limit of the invariant
d
measure for 3 → Z was taken afterwards. Here we present a new point of view in
which a natural Banach space and transfer operators are defined for the infinite lattice of
weakly coupled analytic maps (§1). The space contains consistent families of analytic
d
densities over finite subsets of Z . We take a weighted sup-norm so that the sup-norms
of the densities for the subsystems over finitely many (say N) lattice points is bounded
exponentially in N (§2). We identify an ample subset of this space with a set of rca
(regular, countably additive) measures (§4) that contains the unique invariant probability
density (§2). We derive exponential decay of correlations for this measure and a certain
class of observables from (the proof of) the spectral properties of our transfer operators
(§§2 and 7). The operator for the coupled system and also the invariant measure are (for a
small interaction) in fact perturbations of their counterparts in the uncoupled case. So the
mixing properties are inherited from the single site systems. §8 contains the proofs.
d
Our approach provides a natural setting for an analysis of the full Z Perron–Frobenius
operator in terms of cluster expansions over finite subsets of the lattice. Using residue
calculus we introduce an integral representation for the Perron–Frobenius operator for
finite-dimensional subsystems (§3) which yields a uniform control over the perturbation
and also gives rise to an easy approach to stochastic perturbation (cf. [15]) which, however,
we do not consider here.
Our ‘cluster expansion’ combinatorics (§5) uses ideas from the work of Maes and
Van Moffaert [15] who have introduced a simplified (compared to that in [2]) polymer
expansion. Apart from the analysis of the one-dimensional operator, which is fairly
standard and for which we refer to for example [2], the paper should be self-contained.
1. General setting
d
We consider coupled map lattices in the following setting: the state space is M = (S1)
where S1 = {z ∈ C | |z| = 1} is the unit circle in the complex plane and d a positive
integer.
The map S : M → M is the composition S = F ◦ T ꢀ of a coupling map T ꢀ depending
on a (small) non-negative parameter ꢀ and another parameter for the decay of interaction
(cf. (1)) with an (uncoupled) map F that acts on each component of M separately. We
make the following assumptions.
Assumption 1. F(z) = (fp(zp))p∈ where f : S1 → S1 are real analytic and expanding
d
p
(i.e. fp0 ≥ λ0 > 1) maps that extend for some δ1 holomorphically to the interior of an
def
annulus Aδ = {z ∈ C | −δ1 ≤ ln |z| ≤ δ1} and the family of Perron–Frobenius operators
1
Lf for the single site systems uniformly satisfies a condition specified in §5.1 below (31).
p