ERGODIC PROPERTIES OF QUANTIZED TORAL AUTOMORPHISMS

We study the ergodic properties for a class of quantized toral automor phisms, namely the cat and Kronecker maps. The present work uses and e xtends the results of Klimek and Lesniewski [Ann. Phys. 244, 173-198 ( 1996)]. We show that quantized cat maps are strongly mixing, while Kro necker maps are ergodic and nonmixing. We also study the structure of these quantum maps and show that they are effected by unitary endomorp hisms of a suitable vector bundle over a torus. This allows us to exhi bit explicit relations between our Toeplitz quantization and the semic lassical quantization of cat maps proposed by Hannay and Berry [Physic a D 1, 267-290 (1980)]. (C) 1997 American Institute of Physics.