The geometry of the Weil-Petersson
metric in complex dynamics
In this work, we study an analogue of the Weil-Petersson metric on the space of Blaschke products of degree 2 proposed by McMullen. Via the Bers embedding, one may view the Weil-Petersson metric as a metric on the main cardioid of the Mandelbrot set. We prove that the metric completion attaches the geometrically finite parameters from the Euclidean boundary of the main cardioid and conjecture that this is the entire completion.
For the upper bound, we estimate the intersection of a circle , , with an invariant subset called a half-flower garden, defined in this work. For the lower bound, we use gradients of multipliers of repelling periodic orbits on the unit circle. Finally, utilizing the convergence of Blaschke products to vector fields, we compute the rate at which the Weil-Petersson metric decays along radial degenerations.
- 1 Introduction
- 2 Background in Analysis
- 3 Blaschke Products
- 4 Petals and Flowers
- 5 Quasiconformal Deformations
- 6 Incompleteness: Special Case
- 7 Renewal Theory
- 8 Multipliers of Simple Cycles
- 9 Lower bounds for the Weil-Petersson metric
- 10 Incompleteness: General Case
- 11 Limiting Vector Fields
- 12 Asymptotics of the Weil-Petersson metric
1.1. Basic notation
Let denote the Lebesgue measure on the unit circle , normalized to have total mass 1. Given two points , let denote the hyperbolic distance between and , and be the hyperbolic geodesic connecting and . We use the convention that the hyperbolic metric on the unit disk is while the Kobayashi metric is . For , let . Let be the horoball of Euclidean diameter which rests on and be its boundary horocycle. To compare quantities, we use:
means for some constants ,
1.2. The traditional Weil-Petersson metric
To set the stage, we recall the definition and basic properties of the Weil-Petersson metric on Teichmüller space. Let denote the Teichmüller space of marked Riemann surfaces of genus with punctures. For a Riemann surface , consider the spaces
of holomorphic quadratic differentials with ,
of measurable Beltrami coefficients satisfying
There is a natural pairing between quadratic differentials and Beltrami coefficients given by integration . One has natural identifications
We will discuss two natural metrics on Teichmüller space: the Teichmüller metric and the Weil-Petersson metric. On the cotangent space, the Teichmüller and Weil-Petersson norms are given by
The Teichmüller and Weil-Petersson lengths of tangent vectors are defined by duality, i.e. and . From the definitions, it is clear that the Teichmüller and Weil-Petersson metrics are invariant under the mapping class group . However, unlike the Teichmüller metric, the Weil-Petersson metric is not complete.
For the Teichmüller space of a punctured torus , the mapping class group is . Let us denote the Weil-Petersson metric on by . To describe the metric completion of , we introduce a system of disjoint horoballs. Let denote the horoball that rests on and denote the horoball of Euclidean diameter that rests on . For a fixed , is an -invariant collection of horoballs. When , the horoballs have disjoint interiors but many mutual tangencies. We denote the boundary horocycles by and .
Consider with the usual topology. Extend this topology to by further requiring to be open sets for . Let us also consider a family of incomplete -invariant model metrics on the upper half-plane: for , let be the unique -invariant metric which coincides with the hyperbolic metric on and is equal to on .
For , the metric completion of is homeomorphic to .
Sketch of proof.
To see that the irrational points are infinitely far away in the metric, notice that the horoballs cover the upper half-plane, while by -invariance, the distance between and is bounded below in the metric. Therefore, any path that tends to an irrational number must pass through infinitely many protective shells . In fact, this argument shows that an incomplete path is trapped within some horoball , from which it follows that it must eventually enter arbitrarily small horoballs. By the form of in , it is easy to see that the completion attaches only one point to the cusp at . ∎
Theorem 1.1 (Wolpert).
The Weil-Petersson metric on is comparable to , i.e. for some .
The metric completion of is homeomorphic to .
1.3. Main results
In this paper, we replace the study of Fuchsian groups with complex dynamical systems on the unit disk . Inspired by Sullivan’s dictionary, we are interested in understanding the Weil-Petersson metric on the space
The multiplier at the attracting fixed point gives a holomorphic isomorphism . By putting the attracting fixed point at the origin, we can parametrize by
All degree Blaschke products are quasisymmetrically conjugate to each other on the unit circle, and except for the special map , they are quasiconformally conjugate on the entire disk. For this reason, it is somewhat simpler to work with , the quasiconformal moduli space of a rational map described in [MS]. Given a Blaschke product , an -invariant Beltrami coefficient on the unit disk defines a tangent vector in . Since an -invariant Beltrami coefficient descends to a Beltrami coefficient on the quotient torus of the attracting fixed point, . According to [MS], defines a trivial deformation in if and only if it defines a trivial deformation of . In other words, one has a natural identification of tangent spaces which shows that is the universal cover of .
To make the parallels with Teichmüller theory more explicit, we state our results on the universal cover. For this purpose, we pullback the Weil-Petersson metric on by to obtain a metric on , which we also denote .
The metric on is comparable to on . In particular, the metric completion of is homeomorphic to .
In this paper, we show that is the correct exponent in the conjecture above. More precisely, we show that:
The Weil-Petersson metric on satisfies:
There exists such that on , .
The Weil-Petersson metric on is incomplete. In fact, the Weil-Petersson length of each line segment is finite.
The space naturally embeds into the completion of .
Since the Weil-Petersson metric is a real-analytic metric on , the cusp at infinity in the -model is somewhat special:
Along radial rays , we have a more precise estimate:
Given a rational number , as vertically, the ratio , where is a positive constant independent of .
We conjecture that is a universal constant, independent of .
1.4. Properties of the Weil-Petersson metric
In this section, we give a definition of the Weil-Petersson metric on in the form most useful for our later work. In Section 1.7, we will give equivalent definitions which work on the entire space . For example, the Weil-Petersson metric may be described as the second derivative of the Hausdorff dimension of one-parameter families of Julia sets.
It is convenient to put the Beltrami coefficient on the exterior unit disk. For a Beltrami coefficient , we let denote the reflection of in the unit circle:
Suppose is a Riemann surface and is a Beltrami coefficient. If , we can consider as a -invariant Beltrami coefficient on the unit disk. Let be a solution of . Since the set of all solutions is of the form , the third derivative uniquely depends on . As is an infinitesimal version of the Schwarzian derivative, it is naturally a quadratic differential. In [McM2], McMullen observed that
Similarly, given a Blaschke product , we can solve the equation for . As above, a solution of the equation is well-defined up to adding a holomorphic vector field in so that is uniquely defined. Following [McM2], we define the Weil-Petersson metric provided that the limit exists. In Section 7, we will use renewal theory to establish the existence of this limit for any , invariant under a degree 2 Blaschke product other than .
1.5. A glimpse of incompleteness
We now sketch the proof of the upper bound in Theorem 1.2. To establish the incompleteness of the Weil-Petersson metric, we consider “half-optimal” Beltrami coefficients which take up half of the quotient torus at the attracting fixed point, but are sparse near the unit circle.
The garden is an invariant subset of the unit disk whose quotient is a certain annulus that takes up half of the Euclidean area of the quotient torus. To give an upper bound for the Weil-Petersson metric, we estimate the length of the intersection of with . In general, one has the estimate
In order for this estimate to be efficient, we take to be a collar neighbourhood of the shortest -geodesic in the quotient torus . For the Blaschke product with parameter , , we prove
1.6. A glimpse of the convergence
We now sketch the proof of Theorem 1.3. To understand the behaviour of the Weil-Petersson metric as radially, we study the convergence of Blaschke products to vector fields. For example, as along the real axis, we will see that even though the maps tend pointwise to the identity, their long-term dynamics tends to the flow of the holomorphic vector field . For the radial approach , the maps converge pointwise to a rotation, and therefore the -th iterates tend to the identity. We are thus led to extract a limiting vector field by considering limits of the high iterates of . It turns out that the vector field is a -fold cover of the vector field . In particular, it is independent of .
From the convergence of Blaschke products to vector fields, it follows that the flowers that make up the gardens for have nearly the same shape, up to affine scaling. Intuitively, for the integral average (1.4) to exist, when we replace by say, we expect to intersect twice as many flowers to replenish the integral, i.e. we expect the number of flowers to be inversely proportional in . This leads us to explore an orbit counting problem for Blaschke products. The decay rate of the Weil-Petersson metric is governed by the dependence of the flower count on the parameter variable .
1.7. Notes and references
In this section, we describe the space of Blaschke products of higher degree and equivalent definitions of the Weil-Petersson metric.
Blaschke products of higher degree. More generally, we can consider the space of marked Blaschke products of degree which have an attracting fixed point modulo conformal conjugacy. By moving the attracting fixed point to the origin as before, one can parametrize by
Let denote the multiplier of the attracting fixed point. It is because the maps are marked that we can distinguish the conformal conjugacy classes of and . See [McM3] for more on markings.
Mating. It is a remarkable fact that given two Blaschke products of the same degree, one can find a rational map – the mating of – whose Julia set is a quasicircle which separates the Riemann sphere into two domains such that on one side is conformally conjugate to , and to on the other. The mating is unique up to conjugation by a Möbius transformation. One can prove the existence of a mating by quasiconformal surgery (see [Mil] for details). The mating varies holomorphically with parameters. A natural way to put a complex structure on is via the Bers embedding which takes a Blaschke product and mates it with to obtain a polynomial of degree . Here, the space is considered modulo affine conjugacy. The image of the Bers embedding is the generalized main cardioid in .
For , what is the completion of with respect to the Weil-Petersson metric? Are the additional points precisely the geometrically finite parameters on the boundary of the generalized main cardioid? What is the topology on ?
Wolpert showed that the metric completion of is the augmented Teichmüller space , the action of the mapping class group extends isometrically to and the quotient is the Deligne-Mumford compactification. See [Wol] for more information.
Equivalent definitions of the Weil-Petersson metric. Suppose and is a smooth path with , representing a tangent vector in . Consider the vector field where is the conformal conjugacy between and . If is a Blaschke product other than , one can define by the integral average (1.4), while if , one can use a more complicated integral average described in [McM2].
The definition of the Weil-Petersson metric via mating is slightly more general than the one via quasiconformal conjugacy given earlier because quasiconformal deformations do not exhaust the entire tangent space at the special parameters that have critical relations.
In [McM2], McMullen showed that
is the Julia set of ,
is the conjugacy between and on the unit circle,
is the push-forward of the Lebesgue measure,
is the Lyapunov exponent,
denotes the “asymptotic variance” in the context of dynamical systems.
Since is a Jordan curve, , so and . Similarly, since is a measure supported on the unit circle, , and .
1.8. Relations with quasiconformal geometry
Theorem 1.4 (Smirnov [S]).
For a -quasiconformal map ,
If the dilatation is supported on the exterior unit disk, one has the stronger estimate where .
Theorem 1.5 (Prause, Smirnov [PrSm]).
For a -quasiconformal map , symmetric with respect to the unit circle, one has .
The Weil-Petersson metric on is bounded above by .
For a map , the Bers embedding gives a holomorphic motion of the exterior unit disk given by . Note that the motion is centered at since is the identity. By the -lemma (e.g. see [AIM, Theorem 12.3.2]), one can extend to a holomorphic motion of the Riemann sphere satisfying . Observe that as , . Since each map is conformal on , by the remark following Theorem 1.4, we have as desired. ∎
I would like to express my deepest gratitude to Curtis T. McMullen for his time, energy and invaluable insights. I also want to thank Ilia Binder for many interesting conversations.
2. Background in Analysis
In this section, we explain how to bound the integral (1.4) in terms of the density of the support of . We also discuss a version of Koebe’s distortion theorem for maps that preserve the unit circle.
2.1. Teichmüller theory in the disk
For a Beltrami coefficient , let be a solution of the equation . The following formula is well-known (e.g. see [IT, Theorem 4.37]):
For a Beltrami coefficient and a Möbius transformation , we have whenever . In particular, if is supported on the exterior of the unit disk and , then
The first statement follows from a change of variables and the identity
while the second statement follows from the fact that for all . ∎
To obtain upper bounds for the Weil-Petersson metric, we will use the following estimate:
Suppose is a Beltrami coefficient which is supported on the exterior of the unit disk and has . Then,
Suppose is a Beltrami coefficient which is supported on the exterior of the unit disk and has . Let be its reflection in the unit circle. Then,
is uniformly continuous in the hyperbolic metric.
By the Möbius invariance of , it suffices to prove these assertions at the origin. Clearly,
Hence . This proves . For , recall that . Then,
For , it suffices to observe that the kernel is uniformly continuous at for . ∎
2.2. A distortion theorem
The classical version of Koebe’s distortion theorem says that if is univalent, then for . We will mostly use a version of Koebe’s distortion theorem for maps which preserve the real line or the unit circle:
Suppose is a univalent function which satisfies , and takes real values on . For , is nearly an isometry in the hyperbolic metric on , i.e. .
In particular, distorts hyperbolic distance and area by a small amount:
For , .
If is a round ball contained in , then
Above, “” denotes that . For a set , we call a set of the form a -nearly-affine copy of .
Suppose is a Beltrami coefficient supported on the upper half-ball . It is easy to see that for , where . In terms of quadratic differentials, we have:
On the lower half-ball ,
for some function satisfying as .
2.3. Applications to Blaschke products.
For a Blaschke product , let where ranges over the critical points of that lie inside the unit disk. By the Schwarz lemma, the post-critical set of is contained in the union of and its reflection in the unit circle.
If , the ball is disjoint from the post-critical set, and therefore all possible inverse branches are well-defined univalent functions on . For , let . For Blaschke products, we have the following analogue of Lemma 2.2:
If is an invariant Beltrami coefficient supported on the exterior unit disk, and if the orbit is contained in some with sufficiently small, then
for some function satisfying as .
3. Blaschke Products
In this section, we give background information on Blaschke products. We discuss the quotient torus at the attracting fixed point and special repelling periodic orbits called “simple cycles” on the unit circle. In the next section, we will examine the interface between these two objects.
3.1. Attracting tori
The dynamics of forward orbits of a Blaschke product
is very simple: all points in the unit disk are attracted to the origin. In this paper, we mostly assume that the multiplier of the attracting fixed point . In this case, the linearizing coordinate conjugates to multiplication by , i.e.
It is well-known that (3.2) determines uniquely with the normalization .
Let denote the unit disk with the grand orbits of the attracting fixed and critical point removed. From the existence of the linearizing coordinate, it is easy to see that the quotient is a torus with one puncture. We denote the underlying closed torus by . We will also consider the intermediate covering map defined implicitly by .
For a Blaschke product with , the quotient torus has at most punctures but there could be less if there are critical relations. The reader may view the space consisting of Blaschke products for which as a natural generalization of .
3.2. Multipliers of simple cycles
On the unit circle, a Blaschke product has many repelling periodic orbits or cycles. Since all Blaschke products of degree 2 are quasisymmetrically conjugate on the unit circle, we can label the periodic orbits of by the corresponding periodic orbits of .
A cycle is simple if preserves its cyclic ordering. In this case, we say that has rotation number if . (For simple cycles, we prefer to index the points in counter-clockwise order, rather than by their dynamical order.)
Examples of cycles of degree 2 Blaschke products:
has rotation number 1/2,
has rotation number 1/3,
is not simple.
In degree 2, for every fraction , there is a unique simple cycle of rotation number . We denote its multiplier by . Since Blaschke products preserve the unit circle, is a positive real number (greater than 1). It is sometimes more convenient to work with which is an analogue of the length of a closed geodesic of a hyperbolic Riemann surface.
4. Petals and Flowers
In this section, we give an overview of petals, flowers and gardens. As suggested by the terminology, gardens are made of flowers, and flowers are made of petals. We first give a general definition of a garden, but then we specify to “half-flower gardens” which will be used throughout this work.
In fact, for a Blaschke product , we will construct infinitely many half-flower gardens – one for every outgoing homotopy class of simple closed curves . However, in practice, we use the garden associated to the shortest geodesic in the flat metric on the torus. For parameters , the shortest curve is uniquely defined and has rotation number . It is precisely for this choice of half-flower garden that the estimate (1.6) holds. For example, to study radial degenerations with , we consider gardens where flowers have only one petal (see Figure 2), while for other parameters, it is more natural to use gardens where the flowers have more petals (see Figure 5 below).