Bounds on Eigenfunctions of Quantum Cat Maps

We study ${\mathrm{\ell }}^{\mathrm{\infty }}$ norms of ${\mathrm{\ell }}^2$-normalized eigenfunctions of quantum cat maps. For maps with short quantum periods (constructed by Bonechi and de Bièvre), we show that there exists a sequence of eigenfunctions $u$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\gtrsim (\log N{)}^{-1\mathrm{/}2}$. For general eigenfunctions we show the upper bound $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\lesssim (\log N{)}^{-1\mathrm{/}2}$. Here the semiclassical parameter is $h=(2\pi N{)}^{-1}$. Our upper bound is analogous to the one proved by Bérard for compact Riemannian manifolds without conjugate points.