Translation Invariant Julia Sets

Abstract:  . We show that if the Julia set J(f) of a rational function f is invariant under translation by one and infinity is a periodic or preperiodic point for f , then J(f) must either be a line or the Riemann sphere. 1. Introduction One of the more attractive aspects of Julia sets for rational functions is the display of self-similarity on many different scales. On a global scale, Beardon classified in [2] the finite order rotational symmetries of Julia sets of polynomials. An earlier result along these lines was presented by Baker and Eremenko in [1]. The question arises of what kind of global symmetries the Julia set of a general rational function may possess, i.e., if we assume OE is a Mobius, or linear fractional, transformation such that OE(J(f)) = J(f), and hence OE(F (f)) = F (f) where J(f) is the Julia set of f and F (f) is the Fatou set of f , what can we conclude about f? There are examples of rational functions f with J(f) = C (see [3], x4.3) which is invariant under all Mobius ...