Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

Abstract. We study extreme values of desymmetrized eigenfunctions (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 uniformly 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 ≪ǫ N ǫ for all ǫ > 0 in the case of N square free.