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Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework

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  • It is well-known that the Ginzburg-Landau equation on $R$ has a global attractor [15] that attracts in $L^\infty_{l o c}(R)$ all the trajectories. This attractor contains bounded trajectories that are analytical functions in space. A famous theorem due to P. Collet and JP. Eckmann asserts that the $\varepsilon$-entropy per unit length in $L^\infty$ of this global attractor is finite and is smaller than the corresponding complexity for the space of functions which are analytical in a strip. This means that the global attractor is flatter than expected. We explain in this article how to establish the Collet-Eckmann Theorem in a Hilbert space framework.
    Mathematics Subject Classification: Primary: 35Q56, 35B41, 37L30.

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