V. Baladi, M. Degli Esposti, S. Isola,, E. Jarvenpaa, and A. Kupiainen
The spectrum of weakly coupled map lattices
(482K, postscript)
ABSTRACT. We consider weakly coupled analytic expanding circle maps
on the lattice Z^d (for d >0 ), with small coupling strength
epsilon and coupling between two sites decaying exponentially
with the distance. We study the spectrum of the associated
transfer operators. We give a Frechet space on which the
operator associated to the full system has a simple eigenvalue
at 1 (corresponding to the SRB measure m_epsilon
previously obtained by Bricmont and Kupiainen) and the
rest of the spectrum, except maybe for continuous spectrum,
is inside a disc of radius smaller than one.
For d=1 we also construct Banach spaces of densities
with respect to m_epsilon on which perturbation theory,
applied to the difference of fixed high iterates of the
normalised coupled and uncoupled transfer operators,
yields localisation of the full spectrum of the coupled operator
(i.e., the first spectral gap and beyond).
As a side-effect, we show that the whole spectra of the truncated
coupled transfer operators (on bounded analytic functions) are
O(epsilon)-close to the truncated uncoupled spectra, uniformly
in the spatial size. Our method uses polymer expansions and also
gives the exponential decay of time-correlations for a larger
class of observables than those previously considered.