BOUNDS ON SUPREMUM NORMS FOR HECKE EIGENFUNCTIONS OF QUANTIZED CAT MAPS

Abstract. We study extreme values of desymmetrized eigenfunc-tions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uni-formly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are N for all > 0 in the case of N square free. 1