Symmetries of Julia sets for rational maps

By Gustavo Rodrigues Ferreira

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 .

Lemma 2.1.

is a closed set.

Proof.

If , then every isometry of the Riemann sphere preserves . Hence, is , which is a closed group, and the conclusion follows.

If , we cannot have – since acts transitively on the sphere, it would always be possible to map a point in the Julia set to some point outside the Julia set! Thus, we take . We know that there exists some such that , which is an open set. Therefore, there must be a neighbourhood of that does not intersect . Since is a homogeneous space, this yields a neighbourhood of the identity such that for every . The construction of this neighbourhood is as follows: for every , the homogeneity of implies the existence of some such that . Since the action of on is smooth, the collection of such for every yields a neighbourhood of the identity – which is also in this collection for . By continuity of the group operations, is a neighbourhood of which does not intersect , and thus is open.

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 .

Theorem 2.2.

For a rational map , is (isomorphic to) one of:

(i)

The trivial group;

(ii)

A group of roots of unity;

(iii)

A dihedral group generated by a root of unity and an inversion ;

(iv)

The orientation-preserving symmetries of a regular tetrahedron, octahedron or icosahedron;

(v)

– i.e., the group of isometries of the form and for any ;

(vi)

All isometries of the Riemann sphere.

Proof.

We note that there is little to be done in cases (i) and (vi) from a symmetry point of view. Although their dynamics may be interesting (case (vi), for instance, contains the Lattès maps), we assume now that is neither trivial nor all of .

The first distinction we must make is between a discrete and a continuous symmetry group. In the former, must be a discrete Lie group, and – since is compact – this implies that it is finite. The classification of finite subgroups of in Reference 5, Theorem 4.1 then gives us cases (ii) through (iv). We do remark that we have changed the nomenclature; Carne refers to the roots of unity as the symmetries of a cone on a regular plane polygon, and to the dihedral groups as symmetries of a double cone on a regular plane polygon.

For continuous symmetry groups, we must study the Lie subgroups of . Take, then, the connected component of containing the identity – which must be a Lie subgroup of with an associated Lie subalgebra . Since , where the Lie algebra structure is given by the vector product, it follows that the only proper non-trivial Lie subalgebras of are one-dimensional, and thus is a one-dimensional Lie subgroup of .

Now, every one-dimensional Lie group admits a parametrisation using the Lie exponential. Therefore, we write for some in its Lie algebra . The action of on the Riemann sphere yields a flow defined as , which has as associated vector field – called the action field of – given by

By the hairy ball theorem, there exists such that ; this point satisfies for all , and so it is fixed by every . As every isometry of has exactly two antipodal fixed points, it follows that ’s antipode is also fixed by every element of . Conjugating the Riemann sphere by an isometry such that , we conclude that is conjugate to .

Now, take a that is not fixed by the action of . Its orbit must be a circle, and so is a collection of circles – either a single circle or an uncountable amount of them. In the former case, are the symmetries of a circle, so its elements are either of the form or (furthermore, Eremenko and van Strien Reference 8 showed that either or must be a Blaschke product). We show that the latter case is not possible. The smoothness of each connected component of implies that all multipliers of repelling periodic orbits are real, by a classic argument going back to Fatou (see Reference 17, Corollary 8.11 or Reference 9, Section 46). Now, as shown by Eremenko and van Strien Reference 8, Theorem 1, this implies that either is a Lattès map or is contained in a single circle. We have assumed that , and so we must have contained in a single circle – and therefore equal to a single circle. We fall back to the previous case, and we are done.

Remark 2.3.

It should be noted that all cases in Theorem 2.2 can actually be realised as the symmetry group of some rational map. Case (i), in fact, is shown in Theorem 2.12 to be the most common. For cases (ii) through (iv), consider the group . By Proposition 2.4, any that is also an isometry of belongs to . Now, by a theorem of Doyle and McMullen Reference 7, for any finite subgroup there exists a rational map such that . Thus, by choosing as one of the groups in (ii) - (iv), we can guarantee the existence of rational functions with any finite symmetry group. For the continuous symmetry groups, case (v) is realised by power maps and case (vi), by the Lattès maps.

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.

Proposition 2.4.

Let be a rational function and , and suppose that is not a Lattès map. If for some , then .

Proof.

We will show that the Fatou set of is invariant under . Take . By the Arzelà-Ascoli theorem, for any there is a neighbourhood of satisfying for every . Now, we consider how behaves at . By induction, our hypothesis implies that there exists for all some (which depends on ) such that . Indeed, for the case , is easily given by as per our hypothesis. Now, for any , , and the induction hypothesis gives us . By using that , we can shift the ’s “one-by-one” to obtain . Therefore, ; since is an isometry of the Riemann sphere, it leaves the diameter of a set unchanged, and thus for every . Since the terms on the right-hand side are limited by , this implies (by the Arzelà-Ascoli theorem) that is a normal family at , and thus . Since is also an isometry, we can apply the same reasoning to conclude that , and so – and thus – is invariant under and .

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.

Lemma 2.5.

Let be a domain on the Riemann sphere and , , …points in . Suppose is a function such that:

(i)

is harmonic on ;

(ii)

As , there exists some such that for every (we say that has a logarithmic pole of order at );

(iii)

As , , except maybe for a set of points in of capacity zero.

Then, can be decomposed as

where is the Green’s function of with pole at .

Proof.

Let

We shall prove that the sequence converges to zero, and this will give us the conclusion. First, the fact that is always non-negative implies that – i.e., is a monotonically decreasing sequence. Next, each is harmonic on with the exception of logarithmic poles at for and goes to zero as approaches the boundary of . Therefore, we have that

which means that the sequence converges point-wise to some function . Since it is the limit of a monotonic sequence of harmonic functions, is itself harmonic throughout , and it also satisfies as . By the maximum modulus principle, and we are done.

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 .

Definition 2.6.

Let be a rational function, and its unique measure of maximal entropy. We define its ergodic potential to be the function given by

where denotes the spherical metric on the Riemann sphere.

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 .

Proposition 2.7.

For a non-exceptional rational map without any parabolic or rotation domains and with , and are invariant under .

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

Lemma 2.8.

Let and be rational maps of degree . Then, is completely invariant under and is completely invariant under . In particular, if , then .

Proof.

If , the complete invariance of the Julia set follows immediately from its definition. To prove the converse, we recall that is characterised as the minimal closed set with more than three points which is completely invariant under , hence . By the symmetry of the hypothesis, we also get that , and so they are equal.

Now, consider . In order to conclude that , we shall prove that is invariant under , and vice versa. Firstly, since is a symmetry of , we have by definition that and so it is clear that . Also, by the minimality of the Julia set, this implies that .

All that is left is to prove that , and we shall do it by contradiction. Suppose, then, that is not backward invariant under (since is surjective, this is actually equivalent to assuming that is not completely invariant). Thus, we can take such that contains at least one point – which, by an abuse of notation, we shall denote by – that is not in . In other words, . However, we already know that is a subset of , which is completely invariant under , and therefore . This will be the basis for obtaining a contradiction.

Since is in the Fatou set of , it follows from the Arzelà-Ascoli theorem that is equicontinuous there. Thus, for any , we can take a neighbourhood of such that

By taking a term from the family of , this becomes

Next, we consider what the sequence of mappings means for the diameter. is an isometry of the Riemann sphere; therefore, none of the terms in this expression have any effect on diameter. This means that the end result of is ultimately determined by the iterations of . Also, since and is completely invariant under both and , this means that is always mapped to a neighbourhood of a point in . Since eventually expands all neighbourhoods of points in its Julia set, we can conclude that should eventually grow larger than any small value of , and so we have reached a contradiction.

Proof of Proposition 2.7.

By Lemma 2.8, implies that . This means that , and thus, since the measure of maximal entropy is invariant, we get

In other words, is invariant under symmetries of the Julia set. Now, the expression for becomes