Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps

Details Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps Bounds on supremum norms for Hecke eigenfunctions of quantized cat maps

2006-01-18

arxiv.org/abs/math/0601422v2

Mathematics

Number Theory

16 pages. Introduction expanded; comparison with supremum norms of eigenfunctions of the Laplacian added. Bound for square fre

Scientific paper

10.1007/s00023-006-0300-x

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 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 << N^epsilon for all epsilon>0 in the case of N square free.

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