A combinatorial model for reversible rational maps over finite fields

Abstract. We study time-reversal symmetry in dynamical systems with finite phase space, with applications to birational maps reduced over finite fields. For a polynomial automor-phism with a single family of reversing symmetries, a universal (i.e., map-independent) distri-bution function R(x) = 1−e−x(1+x) has been conjectured to exist, for the normalized cycle lengths of the reduced map in the large field limit [20]. We show that these statistics corre-spond to those of a composition of two random involutions, having an appropriate number of fixed points. This model also explains the experimental observation that, asymptotically, almost all cycles are symmetrical, and that the probability of occurrence of repeated periods is governed by a Poisson law. 1