Applied Maths Seminar: Roberts -- Universal Period Distribution for Reversible Rational Maps over Finite Fields

John Roberts
School of Mathematics and Statistics, University of New South Wales

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.