A NONCOMMUTATIVE VERSION OF THE ARNOLD CAT MAP

We provide a treatment of the ergodic properties of a noncommutative algebraic analogue of the dynamical system known as the Arnold 'cat map' of the two-dimensional torus. Here, the algebra of functions on the torus is replaced by its noncommutative analogue, formulated by Connes and Rieffel, which arises in the quantum Hall effect. Our main results are that (a) the system is mixing and, as in the classical case, the unitary operator, representing its dynamical map, has countable Lebesgue spectrum; (b) for rational values of the noncommutativity parameter, theta, the model is a K-system, in the algebraic sense of Emch, Narnhofer, and Thirring, though not in the entropic sense of Narnhofer and Thirring; (c) for irrational values of theta, except possibly for a set of zero Lebesgue measures, it is neither an algebraic nor an entropic K-system.