Eigenvalue spacings for quantized cat maps

According to one of the basic conjectures in quantum chaos, the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of the typical member of the appropriate ensemble of random matrices. We study one of the simplest examples of this phenomenon in the context of ergodic actions of groups generated by several linear toral automorphisms—'cat maps'. Our numerical experiments indicate that for 'generic' choices of cat maps, the unfolded consecutive spacing distribution in the irreducible components of the N th quantization (given by the N -dimensional Weil representation) approaches the GOE/GSE law of random matrix theory. For certain special 'arithmetic' transformations, related to the Ramanujan graphs of Lubotzky, Phillips and Sarnak, the experiments indicate that the unfolded consecutive spacing distribution follows Poisson statistics; we provide a sharp estimate in that direction.