Bounds on eigenfunctions of quantum cat maps

Abstract

We study ℓ ∞ norms of ℓ 2-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Biévre in F Bonechi and S De Bièvre (2000, Communications in Mathematical Physics, 211, 659–686)) we show that there exists a sequence of eigenfunctions u with . For general eigenfunctions we show the upper bound . Here the semiclassical parameter is . Our upper bound is analogous to the one proved by Bérard in P Bérard (1977, Mathematische Zeitschrift, 155, 249-276) for compact Riemannian manifolds without conjugate points.