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.