John Roberts School of Mathematics and Statistics, University of New South Wales Universal Period Distribution for Reversible Rational Maps over Finite Fields Wednesday 4th March 14:05-14:55pm, Eastern Avenue Lecture Theatre. In a program joint with F Vivaldi (London), we have shown that structural properties of discrete dynamical systems leave a universal signature on the reduced dynamics over finite fields (analogous to the division of quantum spectral statistics into those of certain random matrix ensembles). I will briefly review previous results, then consider reversible rational maps, i.e. those maps in d-dimensional space that can be written as the composition of 2 rational involutions. We study the reduction of such rational maps to finite fields and look to study the proportion of the finite phase space occupied by cycles and by aperiodic orbits and the length distributions of such orbits. We find that the dynamics of these low-complexity highly deterministic maps has some universal (i.e. map-independent) aspects. The distribution is well explained using a combinatoric model that averages over an ensemble of pairs of random involutions in the finite phase space.