ASYMPTOTIC PROPERTIES OF THE PERIODIC-ORBITS OF THE CAT MAPS

We study the periodic orbits of the hyperbolic automorphisms of the unit 2-torus. These are also known as the cat maps, and may be viewed as completely chaotic Hamiltonian dynamical systems. Properties of the orbits of rational points with a given denominator g are considered. Combining results from the theory of numbers with methods from probability theory, we derive an explicit approximation to the distribution of the periods of those orbits which correspond to prime values of g. This result is then used to investigate the asymptotic behaviour of certain properties of the orbits in the limit as g --> infinity.