Extensive Properties of the Complex Ginzburg-Landau Equation

We study the set of solutions of the complex Ginzburg-Landau equation in ${\mathrm{\Re }}^d,d<3$. 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 $N_{Q_L}(ϵ)$ of balls of radius $ϵ$ in $\text{Linfty}(Q_L)$. We show that the Kolmogorov $ϵ$-entropy per unit length, $H_{ϵ}={\lim }_{L\to \mathrm{\infty }}{L}^{-d}\log N_{Q_L}(ϵ)$ exists. In particular, we bound $H_{ϵ}$ by $\text{OO}(\log (1\mathrm{/}ϵ)$, 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: $H_{ϵ}>\text{OO}(\log (1\mathrm{/}ϵ))$