Extensive properties of the complex Ginzburg-Landau equation

We study the set of solutions of the complex Ginzburg-Landau equation in R- d, d < 3. We consider the global attracting set (i.e., the forward map of t he 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-QL (epsilon) of balls of radius e psilon in L-infinity(Q(L)). We show that the Kolmogorov epsilon-entropy per unit length, H-epsilon = lim(L-->infinity) L-d log N-QL(epsilon) exists. I n particular, we bound H-epsilon by O (log(1/epsilon)), which shows that th e attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H-epsilon > O (log(1/epsilo n)).