Extensive Properties of the Complex Ginzburg-Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in R, 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 QL (ε) of balls of radius ε in L∞(Q L ). We show that the Kolmogorov ε-entropy per unit length, H ε = lim L→∞ L −d log N QL (ε) exists. In particular, we bound H ε by O ( log(1/ε) ) , which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: H ε > O ( log(1/ε) ) .