Abstract:

The gamma distribution $R(x)=1-exp(-x)(1+x)$ was previously found to describe the distribution of cycle lengths in permutations arising from the reduction to finite fields of rational automorphisms with a single time-reversal symmetry. In an attempt to investigate the minimal randomness needed to obtain $R(x)$ in the reduction of a time-reversible map, we have studied numerically the Casati-Prosen map. This is a two-parameter reversible map of the torus with zero entropy which preserves rational lattices for rational parameter values. We consider the distribution of the periods over prime lattices and its dependence on the parameters of the map. We conjecture that, for a set of rational parameters having full density, the distribution converges asymptotically to the gamma distribution $R(x).$ We show the non-uniformity of this convergence, a key feature being that $R(x)$ emerges as the limit of a sequence of singular distributions on certain lines in the parameter space.
This is joint work with Natascha Neumärker (Bielefeld) and Franco Vivaldi (London).