Ergodic properties of quantized toral automorphisms (Q2785735)

The authors investigate the ergodic properties of two quantized toral automorphisms, namely Arnold's cat map and Kronecker's map. The algebraic Toeplitz quantization scheme is used which was earlier developed by \textit{S. Klimek} and \textit{A. Leśniewski} [Ann. Phys. 244, 173-198 (1996)]. Ergodic properties of both the maps are obtained. While the quantized cat dynamics is ergodic in some sense and strongly mixing the quantized Kronecker dynamics turns out to be ergodic but not mixing. The authors notice that similar results concerning the cat maps had previously been discussed in part within a different quantization scheme by \textit{F. Benatti, H. Narnhofer}, and \textit{G. L. Sewell} [Lett. Math. Phys. 21, No. 2, 157-172 (1991; Zbl 0722.46033)]. NEWLINENEWLINENEWLINEThe structure of the two maps are also investigated. It is shown that they are both effected by unitary endomorphisms of a suitable vector bundle over a torus. These results permit to establish an explicit relation between the quantization scheme of the present article and the semiclassical quantization approach of \textit{J. H. Hannay} and \textit{M. Berry} [Physica D 1, 267-290 (1980)].