Hans Henrik Rugh
Coupled Maps and Analytic Function Spaces
(530K, Postscript)
ABSTRACT. We consider real-analytic couplings of a direct product of uniformly
analytic and expanding circle maps. We show that when the coupling
is sufficiently small the dynamical system carries a natural invariant
measure which is ergodic and for which time correlations decay
exponentially fast. When a spatial decay of the couplings is present
this is reflected in a spatial decay of correlations in the marginal
densities of the invariant measure, e.g.\ polynomial decay may arise
from a polynomial decay of the couplings. The allowable couplings
include sums of pair, or more generally, $n$-point, interactions
whose norms are summable with a small enough sum.
The space of couplings and the observable algebra
are Banach algebras of functions which are analytic
in infinitely many variables. These algebras act in a natural way
on a Banach module of complex measures with analytic marginal
densities. Using a simple re-summation formula we obtain uniform bounds
for a Perron Frobenius operator associated with the coupled map.
We calculate explicit bounds in some examples.