Entropy of semiclassical measures of the Walsh-quantized baker's map

Nalini Anantharaman (1) , Stéphane Nonnenmacher (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 SPhT - Service de Physique Théorique

We study the baker's map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical regime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an "entropic uncertainty principle".

Mots clés: Quantum chaos; semiclassical measures; entropy; entropic uncertainty principle

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Dynamique Chaotique [nlin.CD]; Systèmes dynamiques [math.DS]

Nalini Anantharaman, Stéphane Nonnenmacher. Entropy of semiclassical measures of the Walsh-quantized baker's map. Annales Henri Poincaré, 2007, 8 (1), pp.37-74. ⟨10.1007/s00023-006-0299-z⟩. ⟨hal-00015984v3⟩