Ergodic properties of quantized toral automorphisms

We study the ergodic properties for a class of quantized toral automorphisms, namely the cat and Kronecker maps. The present work uses and extends the results of Klimek and Leśniewski [Ann. Phys. 244, 173–198 (1996)]. We show that quantized cat maps are strongly mixing, while Kronecker maps are ergodic and nonmixing. We also study the structure of these quantum maps and show that they are effected by unitary endomorphisms of a suitable vector bundle over a torus. This allows us to exhibit explicit relations between our Toeplitz quantization and the semiclassical quantization of cat maps proposed by Hannay and Berry [Physica D 1, 267–290 (1980)].