Vincent Ferrari-Dominguez: Entropy lower bound for semiclassical measures of higher dimensional quantum cat maps

Abstract: We investigate the semiclassical measures of quantum cat maps, a quantized version of linear symplectic maps on the torus. These measures capture the behavior in phase-space of the eigenfunctions of the quantized maps in the semiclassical/high-frequency limit. Under a product hypothesis on the dynamics, we show that in any dimension the metric entropy of any semiclassical measures is at least half of the topological entropy, as conjectured by Anantharaman and Nonnemacher. A conditional quantum entropy and a notion of effective Ehrenfest time are introduced to handle the different timescales that appear in higher dimensions.