Completely invariant Julia sets of polynomial semigroups
  Let G be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that G contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which contains at least three points and is completely invariant under each element of G, is the sphere itself. 
Rich Stankewitz (Texas A&M University)