Rational maps with real multipliers

Let f be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever J(f) belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle. MSC classes: 37F10, 30D05. A simple argument of Fatou [4, Section 46] shows that if the Julia set of a rational function is a smooth curve then all periodic orbits on the Julia set have real multipliers, see also [8, Cor. 8.11]. This argument gives the same conclusion if one only assumes that the Julia set is contained in a smooth curve. By a smooth curve we mean a curve that has a tangent at every point. We prove the converse statement: Theorem 1. Let f : C̄→ C̄ be a rational map such that the multiplier of each repelling periodic orbit is real. Then either the Julia set J(f) is contained in a circle or f is a Lattès map. Corollary 1. If the Julia set of a rational function is contained in a smooth curve then it is contained in a circle. In fact, Theorem 1 holds if all repelling periodic points on some relatively open subset of J(f) have real multipliers. It follows that even if a relatively ∗Supported by NSF grant DMS-0555279. †Supported by a Royal Society Leverhulme Trust Senior Research Fellowship.