Reduction, Dynamics, and Julia Sets of Rational Functions Original Research Article

We consider a rational function φ(z)set membership, variantK(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 φ. 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 an empty Julia set.