Some Rational Maps Whose Julia Sets Are Not Locally Connected

We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected. Introduction and motivations The dynamics of a rational map f acting on Ĉ is concentrated on its Julia set which is (by definition) the minimal compact set invariant by f and f−1 containing at least three points. The question of local connectivity of the Julia set is important when one wants to give a model of the dynamics on some part of this limit set (using Carathéodory’s Theorem for instance), since on its complement, the Fatou set, the dynamics is well understood. This question of local connectivity also concerns the limit set of Kleinian groups, which is the minimal compact set invariant by the group and which contains at least three points (in order to consider only non-elementary Kleinian groups). There is a well-known analogy between theory and results for these two dynamical systems on Ĉ = ∂B which is presented in the dictionary of Sullivan (see [Su, McM1]). All known examples of Kleinian groups have locally connected limit sets (provided they are connected) and there is a model for the action of the group on its limit set; see [AnMa, McM2, Min]. On the other hand, there are several examples of non-locally connected Julia sets for polynomials ; for instance Douady, and Sullivan proved (see [Su]) that any polynomial which has a Cremer periodic point has a non-locally connected Julia set ; there are also several examples among infinitely renormalizable polynomials (see [Mi2, So]). This is not the case for rational maps. Indeed, in [Ro] we give a family of rational maps for which all Julia sets are locally connected. This family contains rational maps with a periodic Cremer point but also rational maps which are infinitely renormalizable near a critical point. In both cases, the Julia set of the rational map contains an homeomorphic image of a non-locally connected Julia set of a quadratic polynomial. It is itself locally connected. What happens is that the Julia set is reconnected by the boundary of small Fatou components that insert themselves “between the hairs”. So, two questions arise naturally. First, are the polynomials very particular cases that do not fit in Sullivan’s dictionary? More precisely, if we define genuine rational maps as rational maps that are not topologically conjugated on any neighborhood of the Julia set to a polynomial, the question is : Does there exist genuine Received by the editors May 11, 2005 and, in revised form, April 7, 2006. 2000 Mathematics Subject Classification. Primary 37F50; Secondary 37F10. Research partially supported by the Morningside Center of Mathematics in Beijing. c ©2006 American Mathematical Society Reverts to public domain 28 years from publication