Collet P., Eckmann J.-P.
Extensive Properties of the
Complex Ginzburg-Landau Equation
(272K, postcript)

ABSTRACT.  We study the set of solutions of the complex Ginzburg-Landau equation
in $\real^d$, d<3. We consider the global attracting set i.e., the
forward map of the set of bounded initial data, and 
restrict it to a cube $Q_L$ of side $L$. We cover this set by a
(minimal) number $N_{Q_L}(\epsilon )$ of balls of radius $\epsilon $
in $\Linfty(Q_L)$. We show that the Kolmogorov \epsilon-entropy per
unit length, 
$H_\epsilon =\lim_{L\to\infty} L^{-d} \log N_{Q_L}(\epsilon)$ 
exists. In particular, we bound $H_\epsilon $ by
$\OO(\log(1/\epsilon ))$, 
which shows that the attracting set is {\em smaller} than the set of
bounded analytic functions in a strip. 
We finally give a positive lower bound: 
$H_\epsilon>\OO(\log(1/\epsilon )) $.