### Reduction, Dynamics, and Julia Sets of Rational Functions, by Rob Benedetto

We consider a rational function *f*(*z*) in
*K*(*z*) in one variable defined over an algebraically
closed field *K* which is complete with respect to a valuation
*v*. We study how the reduction (modulo *v*) of such
functions behaves under composition, and in particular under
iteration. We also investigate the relationship between bad reduction
and the Julia set of *f*. In particular, we prove that under
certain conditions, bad reduction is equivalent to having a nonempty
Julia set. We also give several examples of maps not satisfying those
conditions and having both bad reduction and empty Julia set.

Rob Benedetto <bene@bu.edu>