Value Distribution for Eigenfunctions of Desymmetrized Quantum Maps

In the past few years,much attention has been devoted to the behavior of eigenfunctions of classically chaotic quantum systems. One aspect of this topic concerns their value distribution and specifically their extreme values (see [Be], [AS], [S1], [IS], [HeR], [ABST]). Our aim is to explore this topic for one of the best-studied models in quantum chaotic dynamics—the quantized cat map (see [HB]). This is the quantization of a hyperbolic linear map A of the torus. For a brief background about this model, see Section 2. For each integerN ≥ 1 (the inverse Planck constant), let UN(A) denote the quantization ofA as a unitary operator onHN = L(Z/NZ). In our previous paper [KR], it was observed that there are quantum symmetries ofUN(A)—a commutative group of unitary operators that commute with UN(A). We called these Hecke operators, in analogy with the classical theory ofmodular forms.The eigenspaces ofUN(A) thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we called Hecke eigenfunctions. These can be thought of as the eigenfunctions for the desymmetrized quantum map. In [KR] we showed that the Hecke eigenfunctions become uniformly distributed as N → ∞. In this note we investigate suprema and value distributions of the Hecke eigenfunctions. For general N, we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions. For prime values ofN for which A is diagonalizable modulo N (the “split primes” forA),we obtain muchmore refined, optimal results via the modern