{'fraction_non_alphanumeric': 0.0536231884057971, 'fraction_numerical': 0.015942028985507246, 'mean_word_length': 4.8559322033898304, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hausdorff length of level sets for solutions of the Ginzburg-Landau equation

We study the level sets, and in particular zero sets, of solutions of the complex Ginzburg-Landau equation on a region Omega =(0, 1)2 contained in R2 with periodic boundary conditions. We prove that for every non-constant function u0 belonging to the global attractor and every lambda in C, the quantity min(H1(z in Omega :Reu0(z)=Re lambda ), H1(z in Omega :Imu0(z)=Im lambda )) is bounded by a constant depending only on the parameters of the equation; in particular, this provides an upper bound on H1(z in Omega :u0(z)= lambda ). This result generalizes our previous result treating the one-dimensional case.