Large supremum norms and small Shannon entropy for Hecke eigenfunctions of quantized cat maps

TL;DR

This paper investigates the supremum norms and Shannon entropy of eigenfunctions of quantized cat maps, revealing small support, large supremum norms, and specific entropy properties, especially for composite and prime power N.

Contribution

It provides new entropy estimates and explicit supremum norm calculations for eigenfunctions, highlighting their extremal behavior in quantum chaos models.

Findings

Eigenfunctions can have very small support and large supremum norm for composite N.

An entropy estimate is proven, with eigenfunctions achieving equality in this estimate.

Supremum norms for eigenfunctions are limited to at most four distinct values for prime power N.

Abstract

This paper concerns the behavior of eigenfunctions of quantized cat maps and in particular their supremum norm. We observe that for composite integer values of N, the inverse of Planck's constant, some of the desymmetrized eigenfunctions have very small support and hence very large supremum norm. We also prove an entropy estimate and show that our functions satisfy equality in this estimate. In the case when N is a prime power with even exponent we calculate the supremum norm for a large proportion of all desymmetrized eigenfunctions and we find that for a given N there is essentially at most four different values these assume.