 9850 Collet P., Eckmann J.P.
 Extensive Properties of the
Complex GinzburgLandau Equation
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Feb 6, 98

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Abstract. We study the set of solutions of the complex GinzburgLandau 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 \epsilonentropy 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 )) $.
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