# Symmetries of Julia sets for rational maps

## Abstract

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. Here, we offer partial extensions to Beardon’s results on the symmetry group of Julia sets, and discuss them in the context of singularly perturbed maps.

## 1. Introduction

In complex dynamics, the problem of finding maps with the same Julia set goes back to Julia himself Reference 12. During the 1980s and early 1990s, a complete description for the polynomial case was obtained through the efforts of Baker, Eremenko, Beardon, Steinmetz and others Reference 1Reference 2Reference 3Reference 20. The culmination of this work is the following theorem: given any Julia set for a non-exceptional polynomial, there exists a polynomial such that the set of all polynomials with Julia set is given by

where denotes the set of symmetries of – that is, the set of all complex-analytic Euclidean isometries of preserving A rational function is exceptional if it is conformally conjugate to a power map, a Chebyshev polynomial or a Lattès map – any other rational function is .*non-exceptional*. This result highlights the importance of the group of symmetries for the Julia set of polynomials; it completely determines which polynomials share that Julia set.

More specifically, let be a polynomial, and let denote its Julia set. We are interested in the set

of complex-analytic Euclidean isometries of preserving the ,*symmetries* of for it tells us – via ,Equation 1.1 – whether there are any “interesting” polynomials with the same Julia set. It is fortunate, then, that Beardon Reference 2 had already given a complete characterisation of His results are twofold: first, he described all possible structures for . (it is either trivial, or a finite group of rotations about the same point, or the group of all rotations about a point), and then he gave criteria to determine the structure of for any particular polynomial .

The generalisation to rational maps, however, is not completely understood yet. Levin and Przytycki proved in 1997 that – for a large class of rational functions – having the same Julia set is equivalent to having the same measure of maximal entropy Reference 14. However, Ye Reference 21 gives the following example:

which are non-exceptional rational functions with the same measure of maximal entropy but that cannot be written in the form (or vice versa) for any integer and Möbius transformation showing that the classification given by ,Equation 1.1 fails in general. Here, we prove some partial extensions to Beardon’s results on the symmetry group of Julia sets. We apply these results to obtain a complete description of the symmetries for maps of the form previously studied by McMullen, Devaney and others ,Reference 6Reference 16.

## 2. Results

Since rational functions are not holomorphic throughout all of it is natural to consider them as analytic endomorphisms of the Riemann sphere , Therefore, as opposed to Beardon’s set . of isometries of our set of possible symmetries shall be the set of holomorphic isometries of ,

for the spherical metric on the Riemann sphere. This set is isomorphic as a Lie group to – which means that there is a diffeomorphism that respects the group operations – and as such it is compact and connected, but not simply connected. Given a rational function this allows us to put our first restriction on the structure of ,

the symmetries of its Julia set – and the rational equivalent of Beardon’s .

Though simple, this result has crucial consequences. Firstly, as a closed subset of a compact set, we get that is compact; secondly, by Cartan’s closed subgroup theorem, it follows that is a Lie subgroup of – which means that it is an embedded submanifold of Hence, we obtain our first serious restriction on . .

Now, we need conditions that allow us to choose, among all these possible geometries, which one corresponds to a given rational map We offer a sufficient condition and a necessary one; Ye’s example .Equation 1.2 suggests that complete characterisations, like for polynomials, are not possible.

Our necessary condition even allows us to specify a value for in Proposition 2.4, albeit in a very specific situation. However, we shall need some technical results concerning potentials in order to prove it. We refer to Reference 19, Reference 13 and Reference 4 for the necessary concepts.

Next, we would like to prove that symmetries of somehow respect the critical points of which are central to its dynamical behaviour. Indeed, let , stand for the set of critical points of together with their pre-images (we shall call it the pre-critical set):

We shall demonstrate that any symmetry of must, if satisfies adequate conditions, preserve .

First, it is well-known that any rational function admits a unique measure of maximal entropy (this was proved independently by Lyubich Reference 15 and Freire, Lopes, and Mañé Reference 10 in 1983). Then, Levin and Przytycki showed that, for non-exceptional rational functions without parabolic domains, Siegel discs, or Herman rings (the latter two being called rotation domains), having the same measure of maximal entropy is in fact equivalent to having the same Julia set. We are going to associate a potential to this fundamental measure of rational maps, denoted its ergodic potential, and show that it is continuous and invariant under symmetries of the Julia set. Through this potential, we shall demonstrate the invariance of a certain pluripotential associated to and show that its local minima coincide with critical points of the lift of , to This, in turn, allows us to conclude that . is invariant under symmetries of .

In potential theory, this would be known as the elliptic potential associated to the measure Since the logarithm is a subharmonic function, . is subharmonic in and Okuyama ,Reference 18 proved that it is actually continuous. He also proved that it satisfies where , is the standard area form on the Riemann sphere and and are the differential operators given by and

To prove Proposition 2.7, we shall need this technical lemma.