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 $S_r = \{z : |z| = r\}$, $r \approx 1$, with an invariant subset $\mathcal G \subset \mathbb{D}$ 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.

To set the stage, we recall the definition and basic properties of the Weil-Petersson metric on Teichmüller space.Let T g,n denote the Teichmüller space of marked Riemann surfaces of genus g with n punctures.For a Riemann surface X ∈ T g,n , let Q(X) be the space of holomorphic quadratic differentials with ´X |q| < ∞ and M (X) be the space of measurable Beltrami coefficients satisfying ||µ|| ∞ < ∞.
From the definitions, it is clear that the Teichmüller and Weil-Petersson metrics are invariant under the mapping class group Mod g,n .However, unlike the Teichmüller metric, the Weil-Petersson metric is not complete.
Lemma 1.1.For α > 0, the metric completion of (H, ρ α ) is homeomorphic to H * .Sketch of proof.To see that the irrational points are infinitely far away in the ρ α metric, notice that the horoballs B p/q (2) cover the upper half-plane, while by SL(2, Z)invariance, the distance between H p/q (2) and H p/q (3) is bounded below in the ρ α metric.Therefore, any path γ that tends to an irrational number must pass through infinitely many protective shells B p/q (3) \ B p/q (2).In fact, this argument shows that an incomplete path γ is trapped within some horoball B p/q (3), from which it follows that it must eventually enter arbitrarily small horoballs.By the form of ρ α in B p/q (1), it is easy to see that the completion attaches only one point to the cusp at p/q.Theorem 1.1 (Wolpert).The Weil-Petersson metric on T 1,1 is comparable to ρ 1/2 , i.e. 1/C ≤ ω T /ρ 1/2 ≤ C for some C > 0.
For background on Teichmüller theory and more information on the Weil-Petersson metric, we refer the reader to the books [Hub], [IT] and [Wol].
1.2.Main results.In this paper, we replace the study of Fuchsian groups with complex dynamical systems on the unit disk D = {z : |z| < 1}.Inspired by Sullivan's dictionary, we are interested in understanding the Weil-Petersson metric on the space B 2 = f : D → D is a proper degree 2 map with an attracting fixed point conjugacy by Aut(D).(1.1) The multiplier at the attracting fixed point a : f → f (p) gives a holomorphic isomorphism B 2 ∼ = D.By putting the attracting fixed point at the origin, we can parametrize B 2 by All degree 2 Blaschke products are quasisymmetrically conjugate to each other on the unit circle, and except for the special map z → z 2 , they are quasiconformally conjugate on the entire disk.For this reason, it is somewhat simpler to work with B × 2 := B 2 \ {z → z 2 }, the quasiconformal moduli space M(f ) of a rational map described in [MS].Given a map f ∈ B × 2 ∼ = D * , an f -invariant Beltrami coefficient on the unit disk µ ∈ M (D) f defines a tangent vector in T f B 2 .Since an f -invariant Beltrami coefficient descends to a Beltrami coefficient on the quotient torus of the attracting fixed point, we have M (D) f ∼ = M (T f ).According to [MS], µ defines a trivial deformation in B × 2 if and only if it defines a trivial deformation of T f ∈ T 1,1 .With this correspondence, T 1,1 is naturally the universal cover of B × 2 .We can pullback the Weil-Petersson metric ω B on B 2 by a(τ ) := e 2πiτ to obtain a metric on T 1,1 ∼ = H, which we also denote ω B .
Corollary.The Weil-Petersson metric on B 2 is incomplete.In fact, the Weil-Petersson length of the line segment e(p/q) • [1 − δ, 1) is finite.
Remark.The cusp at infinity is somewhat special: when Im τ is large, Along radial rays a → e(p/q), we have a more precise estimate: Theorem 1.3.Given a rational number p/q ∈ Q, as τ = p/q + it → p/q vertically, the ratio ω B /ρ 1/4 → C q , where C q is a positive constant independent of p.
Conjecture.We conjecture that C q is a universal constant, independent of q.
In a forthcoming work [Ivr], we will show that the Weil-Petersson metric is asymptotically periodic as a → e(p/q) along a horocycle.The proof combines ideas from the work of Epstein [E] on rescaling limits with parabolic implosion.1.3.Properties of the Weil-Petersson metric.In this section, we give a definition of the Weil-Petersson metric on B × 2 ⊂ B 2 in the form most useful for our later work.In Section 1.6, we will give equivalent definitions which work on the entire space B 2 .For example, we will describe the Weil-Petersson metric as the second derivative of the Hausdorff dimension of certain Julia sets.
It is convenient to put the Beltrami coefficient on the exterior unit disk.For a Beltrami coefficient µ ∈ M (D), we let µ + denote the "reflection" of µ in the unit circle: for z ∈ S 2 \ D.
(1.3) Suppose X ∈ T g,n is a Riemann surface and µ ∈ M (X) is a Beltrami coefficient.If X ∼ = D/Γ, we can consider µ as a Γ-invariant Beltrami coefficient on the unit disk.Let v be a solution of ∂v = µ + .Since the set of all solutions is of the form v +sl(2, C), the third derivative v uniquely depends on µ + .As v is an infinitesimal version of the Schwarzian derivative, it is naturally a quadratic differential.In [McM2], McMullen observed that (1.4) Similarly, given a Blaschke product f ∈ B × 2 , we can solve the equation ∂v = µ + for µ ∈ M (D) f .As above, a solution v of the equation ∂v = µ + is well-defined up to adding a holomorphic vector field in sl(2, C), and so v is uniquely defined.Following [McM2], we define the Weil-Petersson metric ||µ|| 2 WP using the integral average (1.4), provided that the limit exists.In Section 7, we will prove the existence of the limit for all degree 2 Blaschke products other than z → z 2 .1.4.A glimpse of incompleteness.We now give a sketch of 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 µ λ • χ G(fa) which take up half of the quotient torus at the attracting fixed point, but are sparse near the unit circle.The garden G(f a ) ⊂ D is an invariant subset of the unit disk, whose quotient A = G(f a )/f a ⊂ T a is an annulus which takes up half of the Euclidean area of the quotient torus.To give upper bounds for the Weil-Petersson metric, we will estimate the length of the intersection of G(f a ) with S r := {z : |z| = r}.We will show that (1.5) In order for the estimate (1.5) to be efficient, we take A to be a collar neighbourhood of the shortest p/q-geodesic in the quotient torus T f ∈ T 1,1 .To prove part (a) of Theorem 1.2, we will show that for a = e 2πiτ with τ ∈ H p/q (η), Combining (1.5) and (1.6), we see that ω B ≤ Cρ 1/4 on {τ : Im τ < 1} as desired.
Remark.The trick of truncating the support of the Beltrami coefficient can be found in the proof of [McM1,Corollary 1.3].
1.5.A glimpse of the convergence ω B /ρ 1/4 → C q .We now give a sketch of the proof of Theorem 1.3.To understand the behaviour of the Weil-Petersson metric as a → e(p/q) radially, we study the convergence of Blaschke products to vector fields.For example, as a → 1 along the real axis, we will see that while the maps f a (z) = z • z+a 1+az tend pointwise to the identity, the long-term dynamics of f a tends to the flow of the holomorphic vector field For the radial approach a → e(p/q), the maps f a (z) → e(p/q)z converge pointwise to a rotation, and therefore the q-th iterates tend to the identity.As before, one can extract a limiting vector field κ q by taking limits of the high iterates of f •q a .It turns out that the vector field κ q is a q-fold cover of the vector field κ 1 .From the convergence of Blaschke products to vector fields, it follows that the flowers that make up the gardens G(f a ) for a ≈ e(p/q) have nearly the same shape, up to affine scaling.We use this to show that where n * (r, f a ) is the number of flowers that intersect the circle S r .By renewal theory, lim r→1 Remark.Intuitively, for the integral average (1.4) to exist, when we replace r = 1 − δ by r = 1 − δ/2 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 δ.
1.6.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, let B d be the space of marked Blaschke products of degree d which have an attracting fixed point modulo conformal conjugacy.By moving the attracting fixed point to the origin as before, one can parametrize B d by Mating.It is a remarkable fact that given two Blaschke products f a , f b , one can find a rational map f a,b (z) -the mating of f a , f b -whose Julia set is a quasicircle J a,b which separates the Riemann sphere into two domains Ω − , Ω + such that on one side f a,b (z) is conformally conjugate to f a , and to f b 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).Remark.Wolpert showed that the metric completion of (T g,n , ω T ) is the augmented Teichmüller space T g,n , the action of the mapping class group Mod g,n extends isometrically to (T g,n , ω T ) and the quotient M g,n = T g,n / Mod g,n is the Deligne-Mumford compactification.WP by the integral average (1.4), while if f (z) = z d , one can use a more complicated integral average described in [McM2].
Remark.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 T f B d at the special parameters f ∈ B d that have critical relations.
In [McM2], McMullen showed that where J 0,t is the Julia set of f 0,t , H t,t : S 1 → S 1 is the conjugacy between f 0 and f t on the unit circle, (H Quasiconformal geometry.The characterizations (1.8) and (1.9) of the Weil-Petersson metric are reflected in quasiconformal geometry in the duality between quasiconformal expansion and quasisymmetric compression: Theorem 1.4 (Smirnov [S]).For a k-quasiconformal map f : Remark.If the dilatation µ(z) = ∂f ∂f is supported on the exterior unit disk, one has the stronger estimate H. dim f (S 1 ) ≤ 1 + k2 where k = 2 k 1+ k2 .
Theorem 1.5 (Smirnov, Prause [PrSm]).For a k-quasiconformal map f : S 2 → S 2 , symmetric with respect to the unit circle, one has From (1.8) and (1.9), it is easy to deduce weaker forms of the infinitesimal statements of Theorems 1.4 and 1.5 in the dynamical setting, i.e. H. dim f (S 1 ) ≤ 1 + Ck 2 and H. dim f * m ≥ 1 − Ck 2 with a constant C > 1. Conversely, using either Theorem 1.4 or Theorem 1.5, it is easy to see that: Proof.For a map f a ∈ B 2 , the Bers embedding β fa gives a holomorphic motion of the exterior unit disk H : B 2 × (S 2 \ D) → C given by H(b, z) := H b,a (z).Note that the motion H is centered at a since H(a, •) is the identity.By the λ-lemma (e.g.see [AIM,Theorem 12.3.2]),one can extend H to a holomorphic motion H of the Riemann sphere satisfying ||µ The pressure metric.In the context of complex dynamics, the expression appeared in the works [PUZ1], [PUZ2] which is based on the earlier work of Makarov [Mak] on the law of the iterated logarithm of harmonic measure.It was also studied on spaces of metric graphs in [PoSh] and in higher Teichmüller theory in [BCLS].
Why degree 2 ?In this paper, we stick to the degree 2 case for concreteness.Many arguments presented here extend almost verbatim to B d , or even to spaces of infinite degree maps -for example, to spaces of universal covering maps of finite complements (while the forward orbits of these infinite degree maps are very wild near the unit circle, backward iteration is nearly affine).
Some useful notation.Let B p/q (η) be a horoball in the unit disk of Euclidean diameter η/q 2 resting on e(p/q) and H p/q (η) = ∂B p/q (η) be its boundary horocycle.
We use m to denote the Lebesgue measure on the unit circle, normalized to have total mass 1.Given two points z 1 , z 2 ∈ D, let d D (z 1 , z 2 ) = inf ´γ ρ denote the hyperbolic distance between z 1 and z 2 , and [z 1 , z 2 ] be the hyperbolic geodesic connecting z 1 and z 2 .We use the convention that the hyperbolic metric on the unit disk is ρ while the Kobayashi metric is |dz| 1−|z| 2 .For z ∈ C * , we use the notation ẑ := z/|z|.To compare quantities, we use:

Background in Analysis
In this section, we explain how to bound the integral average (1.4) in terms of the density of the support of the Beltrami coefficient.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 µ, let v(z) = v µ (z) be a solution of the equation ∂v = µ.The following formula is well-known (e.g.see [IT,Theorem 4.37]): for z ∈ supp µ.
Lemma 2.1.For a Beltrami coefficient µ and a Möbius transformation γ ∈ Aut(S 2 ), we have v γ * µ (z) = v µ (γz) • γ (z) 2 whenever γz ∈ supp µ.In particular, if µ is supported on the exterior of the unit disk and γ ∈ Aut(D), then Proof.The first statement follows from a change of variables and the identity while the second statement follows from the fact that γ * ρ = ρ for all γ ∈ Aut(D).
To obtain upper bounds for the Weil-Petersson metric, we will use the following estimate: Theorem 2.1.Suppose µ is a Beltrami coefficient with ||µ|| ∞ < 1 supported on the exterior of the unit disk.Then, (2.4) Theorem 2.2.Suppose µ is a Beltrami coefficient with ||µ|| ∞ < 1 supported on the exterior of the unit disk.Let µ − := (1/z) * µ be its reflection in the unit circle.Then, (c) v /ρ 2 is uniformly continuous in the hyperbolic metric.
Proof of Theorem 2.1.
Equation (2.4) follows by combining the above L 1 and L ∞ bounds.
Here "A ≈ B" denotes that |A/B − 1| .For a set E ⊂ B(0, t), we call a set of the form h(E) a t-nearly-affine copy of E.
Proof.Write z = x + iy.By the classical version of Koebe's distortion theorem, we see that |h (x) − 1| t.Applying the classical Koebe's distortion again, but this time centered at x, we obtain h(x + iy) − h(x) ≈ t iy and |h (x + iy) − 1| t.The theorem follows by combining the two observations.For two points z 1 , z 2 ∈ H, let d H (z 1 , z 2 ) := inf ´γ ρ H denote the hyperbolic distance between z 1 and z 2 .We note two useful consequences of Theorem 2.3: Proof.To see this, consider the geodesic γ in H that connects z 1 and z 2 .We partition γ into several pieces: γ n := γ ∩ {w : t/2 n+1 ≤ Im w < t/2 n }.Each γ n consists of at most two geodesic segments of hyperbolic length O(1).By Theorem 2.3, ´h(γn) ρ − ´γn ρ = O(t/2 n ).Summing over n = 0, 1, 2, . . ., we see that The reverse inequality may be obtained by applying this argument to h −1 .
Lemma 2.3.Suppose B is a round ball contained in B(0, t) ∩ H, where t < 1/2.The map h distorts the hyperbolic area of B by a multiplicative factor of at most Remark.In the above lemma, one can replace "hyperbolic area" with "Euclidean area" or "area with respect to the volume form |dz| 2 /y." Suppose µ is a Beltrami coefficient supported on the upper half-ball B(0, 1) ∩ H.It is easy to see that for z ∈ B(0, t) ∩ H, . Slightly less evident is the fact that: Lemma 2.4.On the lower half-ball B(0, t) ∩ H, we have: for some function φ 1 (t) satisfying φ 1 (t) → 0 + as t → 0 + .

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.If the multiplier of the attracting fixed point a = 0, near the origin, the linearizing coordinate ϕ a (z) := lim n→∞ a −n • f n a (z) conjugates f a to multiplication by a.This means that In fact, (3.2) determines ϕ a uniquely with the normalization ϕ a (0) = 1.
Let Ω 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 φa : Ω → T × a := Ω/(f a ) is a torus with one puncture.We denote the underlying closed torus by T a .We will also consider the intermediate covering map Higher degree.For a Blaschke product f a ∈ B d with a = f a (0) = 0, the quotient torus T × a has at most (d − 1) punctures but there could be less if there are critical relations.The space 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 f ∈ B 2 by the corresponding periodic orbits of z → z 2 .
A cycle is simple if f preserves its cyclic ordering.In this case, we say that ξ 1 , ξ 2 , . . ., ξ q has rotation number p/q if f (ξ i ) = ξ i+p (mod q) .(For simple cycles, we prefer to index the points {ξ i } ⊂ S 1 in counter-clockwise order, rather than by their dynamical order.) Examples of cycles of degree 2 Blaschke products: In degree 2, for every fraction p/q ∈ Q/Z, there is a unique simple cycle of rotation number p/q.We denote its multiplier by m p/q := (f •q ) (ξ 1 ) which is a positive real number (greater than 1) since Blaschke products preserve the unit circle.It is sometimes more convenient to work with L p/q := log(f •q ) (ξ 1 ) which is an analogue of the length of a closed geodesic of a hyperbolic Riemann surface.
To give a lower bound for the Weil-Petersson metric in small horoballs B p/q (C small ), we will use the fact that the multiplier of the p/q-cycle changes at a "definite rate" when moving in a direction transverse to the horocycles H p/q (η): Theorem 3.1.There exists a constant C small > 0 such that for a Blaschke product f a ∈ B 2 with a ∈ H p/q (η) and η < C small , we have The proof of Theorem 3.1 is based on the argument in [McM4,Theorem 6.1].The main idea is to compare the "petal correspondence" with the holomorphic index formula.The proof will be presented in Section 9.

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 gardens, but then we specify to "half-flower gardens" which will be used throughout this work.
In fact, for a Blaschke product f a ∈ B × 2 , we will construct infinitely many halfflower gardens G [γ] (f a ) -one for every outgoing homotopy class of simple closed curves [γ] ∈ π 1 (T a , * ).However, in practice, we use the garden G(f a ) := G [γ] (f a ) associated to the shortest geodesic γ in the flat metric on the torus.For parameters a ∈ B p/q (C small ), the shortest curve γ is uniquely defined and has rotation number p/q.It is precisely for this choice of half-flower garden that the estimate (1.6) holds.For example, to study radial degenerations with a → 1, 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 6   4.1.Curves on the quotient torus.Inside the first homotopy group π 1 (T a , * ) ∼ = Z ⊕ Z, there is a canonical generator α which is represented by counter-clockwise loops φa ({z : |z| = }) for sufficiently small.By a neutral curve, we mean a curve whose homotopy class in π 1 (T a , * ) is an integral power of α.All non-neutral curves can be classified as either incoming or outgoing , depending on their orientation: a curve γ : R/Z → T a is outgoing if some (and hence every) lift γ for some q ≥ 1.
In other words, γ is outgoing if A complementary (outgoing) generator β is only canonically defined up to an integer multiple of α.In terms of the basis {α, β}, we say that an outgoing curve homotopic to (q − p)α + pβ has rotation number p/q.If we don't specify the choice of β, then p/q is only well-defined modulo 1. 4.2.Lifting outgoing curves.Suppose γ is a simple closed outgoing curve in T × a of rotation number p/q mod 1.It has q lifts to C * under the projection π a : C * → T a , which we denote γ * 1 , γ * 2 , . . ., γ * q .The curves γ * i are "spirals" that join 0 to ∞.Each individual spiral is invariant under multiplication by a q .We typically index the spirals so that multiplication by a sends γ * i to γ * i+p .Let γi := ϕ −1 a (γ * i ) be (further) lifts in the unit disk emanating from the attracting fixed point.
Lemma 4.1.Suppose γ is a simple closed outgoing curve in T × a of rotation number p/q.Then, γi joins the attracting fixed point at the origin to a repelling periodic point ξ i ∈ S 1 of rotation number number p/q.
Proof.Pick a point z 1 on γi , and approximate γi by the backwards orbit of f Since the Blaschke product is asymptotically affine, the hyperbolic distance d D (z n , z n+1 ) between successive points is bounded as it cannot substantially grow for n ≥ N .The boundedness of the backward jumps forces the sequence {z n } to converge to a repelling periodic point ξ i on the unit circle.The same argument shows that the hyperbolic length of the arc of γi from z n to z n+1 is bounded, and therefore γi itself must converge to ξ i .Since f (γ i ) = γi+p , we have f (ξ i ) = ξ i+p .Furthermore, since the lifts γi ⊂ D are disjoint, the points {ξ i } are arranged in counter-clockwise order which means that the repelling periodic orbit ξ 1 , ξ 2 , . . ., ξ q has rotation number p/q.

Definitions of petals and flowers. An annulus
a to an outgoing geodesic of rotation number p/q has q lifts in the unit disk emanating from the origin.We call these lifts petals and denote them P A i , with i = 1, 2, . . ., q.Each petal connects the attracting fixed point to a repelling periodic point.Naturally, the flower is defined as the union of the petals: F = q i=1 P A i .We refer to the attracting fixed point as the center of the flower and to the repelling periodic points as the ends .By construction, flowers are forward-invariant regions.The garden is the totally-invariant region obtained by taking the union of all the repeated pre-images of the flower: We refer to the iterated pre-images of petals and flowers as pre-petals and pre-flowers respectively.In degree 2, a flower has two pre-images: itself and an immediate preflower which we denote F * for convenience.Each pre-flower has two proper preimages.We define the centers and ends of pre-flowers as the pre-images of centers and ends of the flower.We typically label a pre-petal by its end and a pre-flower by its center.

4.4.
Half-flower gardens.We now construct the special gardens that will be used in this work.For this purpose, observe that an outgoing homotopy class [γ] ∈ π 1 (T a , * ) determines a foliation of the quotient torus T a by parallel lines, which are geodesics in the flat metric on T a .Explicitly, we can first foliate the punctured plane C * by the logarithmic spirals and then quotient out by (• a).The branch of log a q is chosen so that π a (γ * θ ) ∈ [γ].Note that since each individual spiral is only invariant under (• a q ), a single line on the quotient torus T a corresponds to q equally-spaced spirals in C * .Therefore, T a is foliated by the parallel lines γ θ := π a (γ * θ ) with 0 ≤ θ < 2π/q.For a Blaschke product f a ∈ B × 2 , the quotient torus T × a has one puncture.Let A 1 = T a \ γ θc be the complement of the "singular line" that passes through this puncture.For 0 < α ≤ 1, let A α ⊂ A 1 be the middle round annulus with Area(A α )/ Area(A 1 ) = α.
By the construction of Section 4.3, the annulus A 1 defines a system of petals P 1 i , i = 1, 2, . . ., q, which we calls whole petals .Similarly, an α-petal P α i is defined as a petal constructed using the annulus A α ⊂ T × a .By default, we take α = 1/2 and write P i = P 1/2 i .We define the half-flower F as the union of all the half-petals.Alternatively, one can describe whole petals and half-petals in terms of linearizing rays.A linearizing ray , or a linearizing spiral if a / ∈ (0, 1), is defined as the preimage γθ := ϕ −1 a (γ * θ ), 0 ≤ θ ≤ 2π emanating from the attracting fixed point.If a whole petal P 1 consists of linearizing rays with arguments in (θ 1 , θ 2 ) = ( x−y 2 , x+y 2 ), then the associated α-petal P α is the union of the linearizing rays with arguments in ( x−αy 2 , x+αy 2 ).Convention.In the rest of the paper, we use this system of flowers.When working with a ≈ e(p/q), we let F = F p/q denote the flower constructed from a foliation of the quotient torus by p/q-curves, arising from the choice of log a q ≈ log 1 = 0.
Higher degree.One can similarly define petals and flowers similarly for Blaschke products of degree d ≥ 3: Call a line γ θ ⊂ T a regular if it is contained in T × a and singular if it passes through a puncture.The singular lines partition T a into annuli, the lifts of which we call whole petals .The number of (p/q)-cycles of whole petals is at most d − 1, but there could be less if several critical points lie on a single line.

Quasiconformal Deformations
In this section, we describe the Teichmüller metric on B × 2 and define pinching deformations.We also define the half-optimal Beltrami coefficients which are supported on the half-flower gardens defined in the previous section.
For a Beltrami coefficient µ with ||µ|| ∞ < 1, let w µ be the quasiconformal map fixing 0, 1, ∞ whose dilatation is µ.Given a rational map f (z) ∈ Rat d , an invariant Beltrami coefficient µ ∈ M (S 2 ) f defines a (possibly trivial) tangent vector in T f Rat d represented by the path For a Beltrami coefficient µ ∈ M (D), one can also consider the symmetrized version w µ which is the quasiconformal map that has dilatation µ on the unit disk and is symmetric with respect to the inversion in the unit circle.For a Blaschke product f ∈ B d and a Beltrami coefficient µ ∈ M (D) f , the symmetric deformation defines a path in B d .Note that while we use symmetric deformations to move around the space B d , we use asymmetric deformations w tµ + • f • (w tµ + ) −1 to compute the Weil-Petersson metric as the definition of ||µ|| WP involves v(z) = d dt t=0 w tµ + (z).The formula for the variation of the multiplier of a fixed point of a rational map will play a fundamental role in this work: Lemma 5.1 (e.g.Theorem 8.3 of [IT]).Suppose f 0 (z) is a rational map with a fixed point at p 0 which is either attracting or repelling.If where T p 0 is the quotient torus at p 0 .The sign is " +" in the repelling case and " −" in the attracting case.

Teichmüller metric.
As noted in the introduction, T 1,1 is the universal cover of B × 2 arising from taking a Blaschke product to its quotient torus T × a ∈ T 1,1 .The Teichmüller metric on B × 2 makes this correspondence a local isometry.More precisely, for a Beltrami coefficient µ ∈ M (D) fa representing a tangent vector in T fa B × 2 , we set (5.2) A well-known result of Royden says that the Teichmüller metric on T 1,1 is equal to the Kobayashi metric; therefore, the Teichmüller metric on B × 2 is half the hyperbolic metric on B × 2 ∼ = D * .(We use the convention that the hyperbolic metric on the unit It is well known that Teichmüller coefficients have the form λ q/|q| with q ∈ Q(T × a ), where Q(T × a ) is the space of integrable holomorphic quadratic differentials on the punctured torus T × a .In particular, this implies that Pulling back to the unit disk, we obtain the Beltrami coefficients µ λ := ϕ * a (µ * λ ) ∈ M (D) fa .We refer to the {µ λ , λ ∈ C} as the optimal Beltrami coefficients.Here, "optimal" is short for "multiplier-optimal" which refers to the fact that µ λ maximizes the absolute value of (d/dt)| t=0 log a t out of all Beltrami coefficients with L ∞ -norm |λ|.
Higher degree.For a Blaschke product f a ∈ B × d of degree d ≥ 3, the quotient torus has d − 1 ≥ 2 punctures, and so Q(T a ) Q(T × a ).Therefore, optimal Beltrami coefficients represent only a complex 1-dimensional set of directions in T T × a T 1,d−1 .Therefore, to understand the Weil-Petersson metric on spaces of Blaschke products of higher degree, one would need to study other types of degenerations.
Given an optimal Beltrami coefficient µ λ and a half-flower garden G(f a ), we define the half-optimal Beltrami coefficient to be µ λ • χ G .Using Lemma 5.1, is easy to see that: Lemma 5.2.The half-optimal Beltrami coefficient µ • χ G is half as effective as the optimal Beltrami coefficient µ, i.e. the map

Pinching deformations.
A closed torus X ∈ T 1 carries a natural flat metric of area 1, arising from the uniformization π : C → X.Given a simple, closed Euclidean geodesic γ ⊂ X ∈ T 1 , the pinching deformation {X t } 0≤t<1 is "the most efficient deformation" that shrinks the Euclidean length of γ.More precisely, It is also useful to define the notion of pinching deformations for annuli: given an annulus A = A 0 , the pinching deformation {A t } t≥0 is the deformation which shrinks the length of the core curve the fastest (alternatively, the modulus of A t grows as quickly as possible).For the annulus A r,R := {z : r < |z| < R}, the pinching deformation is given by the Beltrami coefficients t • µ pinch = t • (w/w) • (dw/dw).It is easy to see that pinching a torus X with respect to an Euclidean geodesic γ is the same as pinching the annulus A = X \ γ.

Incompleteness: Special Case
In this section, we give a simple proof of the incompleteness of the Weil-Petersson metric in B 2 when we take a → 1 along the real axis.Our goal is not to present the most general argument, but to give the fastest route to the result.As noted in the introduction, to show that ω B /ρ D * (1 − |a|) 1/4 on (1/2, 1], it suffices to prove: Theorem 6.1.For a Blaschke product f a ∈ B 2 with a ∈ [1/2, 1), we have (6.1) We will deduce Theorem 6.1 from: Theorem 6.2.For a Blaschke product f a ∈ B 2 with a ∈ [1/2, 1), (a) Every pre-petal lies within a bounded hyperbolic distance of a geodesic segment.
Recall that a horocycle connecting two points is exponentially longer than the geodesic: if −x + iy, x + iy ∈ H, then the hyperbolic length of the horocycle joining them is 2 • x/y while the length of the geodesic joining them is ´π−θ θ dt sin t = 2 log(cot(θ/2)) where cot θ = x/y.As cot θ ≈ 1/θ for θ small, this is approximately 2 log(2 • x/y).With this in mind, we argue as follows: Proof of Theorem 6.1.By part (a) of Theorem 6.2, the hyperbolic length of the intersection of S r with any single pre-petal is O(1).By part (b) of Theorem 6.2, whenever the circle S r intersects a pre-petal, an arc of hyperbolic length O 1 − |a| is disjoint from the other pre-petals.Therefore, only the O 1 − |a| -th part of S r can be covered by pre-petals.
Proof.By symmetry, the linearizing ray γ0 = ϕ −1 a ((0, ∞)) is precisely the line segment (0, 1) which lies within a bounded hyperbolic neighbourhood of a geodesic ray.It remains to show that the petal P(f a ) lies within a bounded hyperbolic neighbourhood of γ0 .Suppose z ∈ P(f a ) lies outside a small ball B(0, δ).Let F be the fundamental domain bounded by {ζ : |ζ| = δ} and its image under f a .Under iteration, z eventually lands in F , e.g.
On the other hand, the limiting argument of the critical point lim n→∞ arg f •n (c) = π since the forward orbit of the critical point is contained in the segment (−1, 0).Therefore, we can pick a point This completes the proof.
6.2.The structure lemma.To establish the quasi-geodesic property for pre-petals, we show the "structure lemma" which says that the pre-petals are nearly-affine copies of the immediate pre-petal, while f : P −1 → P is approximately the involution about the critical point, i.e.
, where m 0→c = z+c 1+cz and Naturally, the petals and pre-petals of f are defined as the images of petals and pre-petals of f under m c→0 .
Lemma 6.2 (Structure lemma).For a ∈ [1/2, 1) on the real axis,  By the Schwarz lemma, given two pre-petals To complete the proof, it suffices to show that pre-petals P ζ 1 and P ζ 2 are far apart in the case that they have a common parent, e.g. when f We prove this using a topological argument.Observe that −1 and 1 separate the unit circle in two arcs, each of which is mapped to S 1 \ {1} by f a .Therefore, any path in the unit disk connecting P ζ 1 and P ζ 2 must intersect the line segment (−1, 1) ⊂ P 1 1 ∪ P 1 −1 .However, we already know that the distance between P ζ i to either P 1 and P −1 is greater than d D (0, a) − O(1) which tells us that the hyperbolic ( 1 2 • d D (0, a) − O(1))neighbourhood of (−1, 1) is disjoint from P ζ 1 and P ζ 2 .This completes the proof.

Renewal Theory
In this section, we show that for a Blaschke product other than z → z d , the integral average (1.4) defining the Weil-Petersson metric converges.The proof is based on renewal theory, which is the study of the distribution of repeated pre-images of a point.In the context of hyperbolic dynamical systems, this has been developed by Lalley [La].We apply his results to Blaschke products, thinking of them as maps from the unit circle to itself.Using an identity for the Green's function, we extend renewal theory to points inside the unit disk.Renewal theory will also be instrumental in giving bounds for the Weil-Petersson metric.
For a point x on the unit circle, let n(x, R) denote the number of repeated preimages y (i.e.f •n (y) = x for some n ≥ 0) for which log |(f •n ) (y)| ≤ R. Also consider the probability measure µ x,R on the unit circle which gives equal mass to each of the n(x, R) pre-images.We show: Furthermore, as R → ∞, the measures µ x,R tend weakly to the Lebesgue measure.
For a point z ∈ D, let N (z, R) be the number of repeated pre-images of z that lie in the ball centered at the origin of hyperbolic radius R.
Theorem 7.2.Under the assumptions of Theorem 7.1, we have As before, when R → ∞, the N (z, R) pre-images become equidistributed on the unit circle with respect to the Lebesgue measure.
7.1.Green's function.Let G(z) = log 1 |z| be the Green's function of the disk with a pole at the origin.It is uniquely characterized by three properties: |z| is harmonic near 0.
Lemma 7.1.For a Blaschke product f ∈ B d , we have To prove Lemma 7.1, it suffices to check that f (w i )=z G(w i ) also satisfies the three properties above.We leave the verification to the reader.From equation ( 7.3), it follows that the Lebesgue measure on the unit circle is invariant under f .Indeed, for a point x ∈ S 1 , one can apply the lemma to z = rx and take r → 1 to obtain f (y)=x |f (y)| −1 = 1.(Alternatively, one can apply ∂ ∂z to both sides of (7.3) to obtain the somewhat stronger statement f (w)=z In fact, the Lebesgue measure is ergodic.The argument is quite simple (see [SS] or [Ha]); for the convenience of the reader, we reproduce it here: given an invariant set E ⊂ S 1 , form the harmonic extension u a harmonic function in the disk which is invariant under f .Since 0 is an attracting fixed point, u E must actually be constant, forcing E to have measure 0 or 1 as desired.From the ergodicity of Lebesgue measure, it follows that conjugacies of distinct Blaschke products are not absolutely continuous.7.2.Weak mixing.For the map z → z d , the pre-images come in packets and so n(x, R) is a step function.Explicitly, While n(x, R) has exponential growth, due to the lack of mixing, some values of R are special.For all other Blaschke products, we do have the required mixing property and Theorem 7.1 follows from [La, Theorem 1 and formula (2.5)].
Sketch of proof of Theorem 7.1.In the language of thermodynamic formalism, we must check that the potential φ f (x) = − log |f (x)| is non-lattice, i.e. that there does not exist a bounded function γ such that φ = ψ + γ − γ • f with ψ valued in a discrete subgroup of R. To the contrary, if such a ψ exists, then the multiplier spectrum {log(f is contained in a discrete subgroup of R. Following the proof of [PP,Proposition 5.2], we see that there exists a function w ∈ C α (Σ) satisfying w(f (x)) = e iaφ f (x) w(x), for some a ∈ R \ {0}. (7.4) Here, Σ = {0, 1, . . ., d − 1} N is the shift space which codes the dynamics of f on the unit circle.However, if we work directly on the unit circle and repeat the proof of [PP,Proposition 4.2], we obtain a function w ∈ C α (S 1 ) satisfying (7.4).Since w(x) is non-vanishing and has constant modulus, we can scale it by a constant if necessary so that |w(x)| = 1.By comparing the topological degrees of both sides of (7.4), we see that the topological degree of w is 0. In particular, w admits a continuous branch of logarithm.
If w(x) = e iv(x) then v • f = a • φ f + v + 2πk for some constant k ∈ Z.Therefore, φ f ∼ 2πk/a is cohomologous to a constant.This tells us that the Lebesgue measure m must also be the measure of maximal entropy.However, the measure of the maximum entropy is a topological invariant, thus if we have a conjugacy h between z d and f (z), then the measure of the maximal entropy is h * m.However, we know that the conjugacies of distinct Blaschke products are not absolutely continuous, therefore, we must have f (z) = z d .7.3.Computation of entropy.Since the dimension of the unit circle is equal to 1, the entropy h(f, m) of the Lebesgue measure coincides with the Lyapunov exponent 1 2π ´log |f (e iθ )|dθ, which we can compute using Jensen's formula: Lemma 7.2.The entropy of the Lebesgue measure for the Blaschke product f a (z) with critical points {c i } and zeros {z i } is given by In particular, for degree 2 Blaschke products, as a tends to the unit circle, the entropy h(f a , m) ∼ 1 − |c| ∼ 2(1 − |a|).7.4.Laminated area.For a measurable set E in the unit disk, let Ê denote its saturation under taking pre-images, i.e.Ê = {ζ : f •n (ζ) ∈ E for some n ≥ 0}.For a saturated set Ê, we define its laminated area as Â( Ê) = lim r→1 − 1 2π |E ∩ S r | and say that "E subtends the Â( Ê)-th part of the lamination."By Koebe's distortion theorem (see Section 2.2), we have the following useful estimate: Proof.By breaking up the set E into little pieces, we may assume that E ⊂ B(x, t) for some x ∈ S 1 .We claim that The claim follows in view of the the identity f the Lebesgue measure is invariant).Therefore, we may assume that E ⊂ U t with t > 0 arbitrarily small, i.e. we can pretend that f −1 is essentially affine.By approximation, it suffices to consider the case when E = R is a "rectangle" of the form with 1 , 2 small.For k large, the circle S 1−δ/k = {z : |z| = 1−δ/k} intersects ≈ 1 k/h pre-images of R. As the hyperbolic length of S 1−δ/k is ∼ 2πk/δ and each pre-image has "horizontal" hyperbolic length of ≈ 2 , the laminated area Â( R) ≈ 1 2 2πh • δ as desired.
Recall from [McM2] that a continuous function h : D → C is almost-invariant if for any > 0, there exists r( ) < 1, so that for any orbit z Theorem 7.3.Suppose f is a Blaschke product other than z → z d , and h is an almost-invariant function.Then the limit lim r→1 − 1 2π ´|z|=r h(z)dθ exists.
Proof.Let E be a backwards fundamental domain near the unit circle, e.g.take E = f −1 (B(0, s)) \ B(0, s) with s ≈ 1. Split E into many pieces on which h is approximately constant.By applying Lemma 7.3 to each piece and summing over the pieces, we see that as r → 1, 1 2π ´|z|=r h(z)dθ oscillates within an arbitrarily small multiplicative factor.Therefore, the limit exists.
Applying the above theorem with h = |v /ρ 2 | 2 , which is almost-invariant by Lemma 2.5, gives: Corollary.Given a Blaschke product f ∈ B d other than z → z d , the limit in the definition of the Weil-Petersson metric (1.4) exists for every vector field v that is associated to a tangent vector T f B d .

Lower bounds for the Weil-Petersson metric
In this section, we explain how to obtain lower bounds for the Weil-Petersson metric using the multipliers of repelling periodic orbits on the unit circle.We first consider the "Teichmüller case" and then handle the "Blaschke case" by linear approximation.However, the approximation argument comes with a price: unlike in the Teichmüller case, to give a lower bound for the Weil-Petersson metric we must insist that the quotient torus of the repelling periodic orbit changes a definite rate in the Teichmüller metric.It is precisely this "minor" detail which prevents us from showing that the completion of the Weil-Petersson metric on B 2 attaches precisely the points e(p/q) ∈ S 1 and forces us to restrict our attention to small horoballs.
For instance, it is well-known that in Teichmüller space, the Weil-Petersson length of a curve X : . However, we are unable to prove the analogous statement for the Weil-Petersson metric on B d where we replace the "length of a hyperbolic geodesic" by "the (logarithm of the) multiplier of a periodic orbit."The difficulty is caused by the error term in Lemma 2.5.For details, see the proof of Theorem 8.1 below.8.1.Lower bounds in Teichmüller space.Consider a linear map f (z) = λz with λ > 1.Given a Beltrami coefficient µ ∈ M (H) f supported on the upper half-plane, form the maps f t = w tµ • f 0 • (w tµ ) −1 .Since we use the asymmetric deformations w tµ , the multipliers λ t = f t (0) are not necessarily real.We view v = (d/dt)| t=0 w tµ as a holomorphic vector field on the lower half-plane.
Theorem 8.1 (Blowing up).Suppose f (z) ∈ B 2 is Blaschke product and f •q (ξ) = ξ is a repelling periodic point on the unit circle with Proof.By the corollary to Lemma 8.2, we can find a small ball B 0 of a definite hyperbolic size near ξ for which Using the forward iteration of f (and Koebe's distortion theorem), we can blow up this ball so that its Euclidean size is comparable to δ c .Note that due to the error term in Lemma 2.5, in order for the estimate (8.8) to remain meaningful, we must insist that | L0,t (ξ)/L(ξ)| 1.
Theorem 8.2 (Blowing down).If additionally the multiplier is bounded from below as well as from above, i.e. if Sketch of proof.In view of Lemma 2.5, the estimate (8.7) holds for the inverse images of B. Since the multiplier is bounded from both above and below, we can choose c 1 and c 2 small enough so that the repeated inverse images of B are disjoint from B (and thus from each other).By Lemma 7.3, the Lebesgue measure of the intersection of B with a circle {z : |z| = r} for r sufficiently close to 1 is bounded from below, which gives (8.9).
In Section 10, we will modify the "blowing up" and "blowing down" techniques to give lower bounds for the Weil-Petersson metric when the multiplier of the repelling periodic orbit is small.Remark.To give lower bounds for the Weil-Petersson metric, we used the gradient of the multiplier of a periodic orbit in the µ direction.In view of the the identities we can also use the gradient of the multiplier in the Blaschke slice, i.e. in the µ + µ − or iµ + (iµ) − directions.

Multipliers of Simple Cycles
In this section, we study the behaviour of repelling periodic orbits on the unit circle with small multipliers.In particular, we prove Theorem 3.1.We first make some useful definitions.Let T p/q denote the quotient torus associated to the repelling periodic orbit of rotation number p/q and T in p/q ⊂ T p/q be the half of the torus which is associated to points inside the unit disk.Let P 1 p/q ⊂ T in p/q be the footprint of F 1 in T in p/q , i.e. the part of T in p/q filled by F 1 .The footprint P p/q of F = F 1/2 is defined similarly.To prove Theorem 3.1, we need the following lemma: Lemma 9.1.There exists C small > 0 sufficiently small so that for a ∈ B p/q (C small ), (i) The footprint P 1 p/q of the whole petal contains a definite angle of opening at least 0.99 π. (ii) The footprint P p/q of the half-petal is contained in a central angle of 0.51 π.
To prove Lemma 9.1, we need two preliminary facts: a formula for the conformal modulus of an annulus, and the holomorphic index theorem.We discuss these ideas in the next two sections.9.1.Conformal modulus of an annulus.We use the convention that the annulus A r,R := {z : r < |z| < R} has modulus log(R/r) 2π , which is the extremal length of the curve family Γ ↑ (A r,R ) consisting of curves that join the two boundary components of A r,R .We denote the dual curve family by Γ (A r,R ), consisting of curves that separate the two boundary components.Then, λ Γ ↑ (A) • λ Γ (A) = 1.For background on extremal length and moduli of curve families, we refer the reader to [GM].
If B ⊂ A is an essential sub-annulus of A, we say that B is round in A if the pair (A, B) is conformally equivalent to a pair of concentric round annuli (A r,R , A r ,R ) with 0 < r < r < R < R < ∞.Alternatively, B is round in A if the pinching deformations for A and B are compatible, i.e. if µ pinch (B) = µ pinch (A)| B .
Suppose T * ⊂ C * is a region bounded by two Jordan curves γ 1 , γ 2 which are invariant under multiplication by α, with |α| > 1.By analogy with (9.1), we define the generalized angle β between γ 1 and γ 2 by the formula mod(T * / {z ∼ αz}) = β Re 1 log α .9.2.Holomorphic index formula.We now turn to the holomorphic index formula.If g(z) is a holomorphic map, the index of a fixed point ζ is defined as where γ is any sufficiently small counter-clockwise loop around ζ.If the multiplier λ = g (ζ) is not 1, this expression reduces to 1 1−λ .By the residue theorem, one has: Theorem 9.1 (Holomorphic Index Formula).Suppose R(z) is a rational function and {ζ i } is the set of its fixed points.Then, For a Blaschke product f ∈ B d , the holomorphic index formula says that where the sum ranges over the repelling fixed points on the unit circle, and a = f (0) is the multiplier of the attracting fixed point.9.3.Petal correspondence.Since a whole petal joins the attracting fixed point to a repelling periodic point, it provides a conformal equivalence between an annulus A 1 ⊂ T × a with P 1 p/q ⊂ T p/q .As there are q whole petals at the attracting fixed point, where β is the generalized angle representing the modulus of mod P 1 p/q .Observe that the holomorphic index formula gives a lower bound on m p/q : Proof of Lemma 9.1.Suppose a ∈ H p/q (η).If η > 0 is small, then a q ∈ H 1 ( η+θ q ) with |θ| small.On this horocycle, Re 1 log(1/a q ) ≈ q η+θ while the Poisson kernel 1−|a q | 2 |1−a q | 2 ≈ 2q η+θ .Following [McM4], comparing (9.4) and (9.5), we deduce that the angle β is close to π, when η is small.By the standard modulus estimates (see Lemmas 9.3 and 9.4 below), it follows that the footprint P 1 p/q must contain an angle of opening close to π.They also show that the footprint of the half-petal P p/q is contained in a central angle of opening slightly greater than π/2.We now prove Theorem 3.1: Proof of Theorem 3.1.For (i), we plug in β ≈ π into (9.4) to obtain 1/ log m p/q ≈ 2/η or m p/q ≈ 1 + η/2.Part (ii) is slightly harder.Since the footprint of the whole petal P 1 p/q contains an angle of > 0.51π, it is easy to construct some Beltrami coefficient ν which effectively changes the multiplier of the repelling periodic orbit, i.e. (d/dt)| t=0 m p/q (f tν ) 1.
As B 2 is one-dimensional, we see that for an optimal Beltrami coefficient µ, we must have either This is sufficient for applications to the Weil-Petersson metric; however, for completeness, we will show that the first alternative holds when µ = µ pinch ∈ M (D) is the optimal pinching coefficient built from the attracting torus.
As the dynamics of f •q is approximately linear near a repelling periodic point, µ = µ pinch descends to a Beltrami coefficient ν ∈ M (T p/q ), with supp ν ⊂ T in p/q .Since µ| A 1 is the optimal pinching coefficient for A 1 , ν| P 1 p/q is the optimal pinching coefficient for the annulus P 1 p/q .By Lemma 9.1, when η > 0 is small, the footprint P 1 p/q takes up most of T in p/q , and since T in p/q is a round annulus in T p/q , ν is approximately equal to the optimal pinching coefficient for T p/q on T in p/q .When we consider deformations f tµ in the Blaschke slice, we use the Beltrami coefficient µ + µ + , which corresponds to ν + ν + ∈ M (T p/q ).As ν + ν + ∈ M (T p/q ) is approximately equal to the optimal pinching coefficient for T p/q (at least away from the trace of the unit circle in T p/q ), it is clear that dm p/q /dη 1 when η is sufficiently small.9.4.Standard modulus estimates.For the convenience of the reader, we state the standard estimates for moduli of annuli that we have used in the proofs of Lemma 9.1 and Theorem 3.1.
Lemma 9.3.Suppose A = A r,R and B ⊂ A is an essential sub-annulus.For any > 0, there exists δ > 0 and m 0 > 0 such that if mod A > m 0 and mod B ≥ (1 − δ) mod A, then B contains the "middle" annulus of modulus (1 − ) mod A.
By symmetry, we may assume that S is missing a curve joining z 1 = iy 0 and z 2 = ( /2)m + iy 1 .Note that m = λ Γ↔(R) is the extremal length of the horizontal curve family.Giving an upper bound on the extremal length of Γ ↔ (S) is equivalent to finding a lower bound on the extremal length of the vertical curve family Γ (S).For this purpose, consider the metric 1− /4 as desired.We can deduce the original statement with annuli from the special case when (AB) = (CD) + i by representing the pair B ⊂ A as A = R/{z ∼ z + i} and B = S/{z ∼ z + i}.Indeed, mod A = m while mod B ≥ mod S can only increase since a path in Γ (B) contains a path in Γ (S).
Essentially the same argument shows that: Lemma 9.4.Suppose A = A r,R has modulus mod A > m 0 and B 1 , B 2 , B 3 ⊂ A are three essential disjoint annuli, with B 2 sandwiched between B 1 and B 3 .For any > 0, there exists δ > 0 and m 0 > 0 such that if mod A > m 0 and then B 2 is contained within the "middle" annulus of modulus (1/2 + ) mod A.
We leave the details to the reader.

Incompleteness: General Case
In this section, we prove Theorem 1.2 which says that the Weil-Petersson metric is comparable to the model metric ρ 1/4 in small horoballs (outside the small horoballs, the upper bound is automatic: see the corollary to Theorem 1.4 or use part (a) of Theorem 2.2).Unraveling definitions, we need to show that if For a ∈ B p/q (C small ), the flowers are still well-separated; however, they no longer satisfy the quasi-geodesic property.Nevertheless, it is still the case that the intersection of G(f a ) with a circle {z : |z| = r} for r close to 1 is small.The following lemma is the key to both the upper and lower bounds for ||µ • χ G || 2 WP : Lemma 10.1.Suppose that ξ 1 , ξ 2 , . . ., ξ q is a repelling periodic orbit of a Blaschke product f ∈ B 2 whose multiplier is m < M small := 1 + 1 16 .There exists a constant K > 0 sufficiently large such that the branch of (f •q ) −1 which takes ξ i to itself, maps B(ξ i , R) strictly inside of itself, where Corollary.For each i = 1, 2, . . ., q, the formula defines a univalent holomorphic function on B(ξ i , R) satisfying By Koebe's distortion theorem, Lemma 10.1 implies that the dynamics of f •q is nearly linear in the balls Remark.Note that Lemma 10.1 is only significant for repelling periodic orbits with small multipliers.For m > M small , one can apply Koebe's distortion theorem to the inverse branch (f •q ) −1 on B(ξ i , δ c ) to see that (f •q ) −1 maps the ball B(ξ i , δ c /K) inside of itself.
Combining Lemma 10.1 with part (ii) of Lemma 9.1 gives: Theorem 10.1 (Flower bounds).There exists a constant K > 0 sufficiently large such that for any f a ∈ B 2 with a ∈ B p/q (C small ), Remark.We do not need to know any information about the behavior of the flower within the ball B(0, 1 − 0.5 • R).
With the help of Theorem 10.1, we extend the flower separation and structure lemmas to the wider class of parameters.Since the statements are interrelated, we state them as a single theorem: Theorem 10.2.For a ∈ H p/q (η) with η < C small , Using Theorems 10.1 and 10.2, it is easy to deduce Theorem 1.2.We give the details in Section 10.3.10.1.Linearization at repelling periodic orbits.To show Lemma 10.1, we recall a formula for the derivative of a Blaschke product on the unit circle: (10.4) In particular, the absolute value of the derivative of a Blaschke product is always greater than 1 on the unit circle.Specifying Lemma 10.2 to degree 2 and rearranging, we obtain: Lemma 10.4.There exists a constant K > 0 such that for any degree 2 Blaschke product In view of (10.5), this gives Therefore, (10.6) holds with K = min(1/3, 1/(2C)).
Proof of Lemma 10.1.
, by Lemma 10.4, f −1 is a contraction on each ball B(ξ i , R).Therefore, the composition (f •q ) −1 is a contraction as well.10.2.Separation and structure revisited.To prove Theorem 10.2, we need a convenient way of estimating hyperbolic distances between points in the unit disk: Lemma 10.5.Suppose z 1 , z 2 ∈ D and z 0 is the point on the hyperbolic geodesic [z 1 , z 2 ] closest to origin.If z 0 does not coincide with either endpoint, then (10.8) Sketch of proof.We prove an analogue of (10.8) in the upper half-plane.Observe that for a point z = x+iy ∈ H, the geodesic distance between z and i|z| is log(|x|/y)+O(1), while the vertical distance is log(|z|/y).If |x| < y, then both distances are O(1), while We leave the original statement as an exercise for the reader. (10.9) To deduce the corollary from lemma, it suffices to observe that for two points We now deduce Theorem 10.2 from Theorem 10.1: Proof of Theorem 10.2.Recall from Theorem 3.1 that η (m − 1).By Theorem 10.1, the flower F is contained in D \ B(ĉ, R/2).This implies that This proves (a) and (a ).We can assume that CKη is small, otherwise the theorem is vacuous.Applying Koebe's distortion theorem to the appropriate branch of f −1 on B(−ĉ, 1), we see that F * is a nearly-affine copy of F. Furthermore, since c is the midpoint of the hyperbolic geodesic [0, −a], we must have F * ≈ − F. Therefore, and This proves (b) and (b ).Finally, (c) follows from the Schwarz lemma and the trick used in the proof of part (b) of Theorem 6.2.10.3.Proof of the main theorem.We are now ready to show that for a ∈ H p/q (η) with η < C small .
We first show the upper bound.Reflecting (10.1) about the critical point, we see that the immediate pre-flower F * is contained in the union of the reflections (10.10)Assuming the claim, Lemma 7.3 tells us that Â(G(f a )) δc as desired.To prove the claim, we need to carefully reflect the flower about the critical point.
The reflection B * of the ball B(0, The total contribution of these sectors to the integral (10.10) is roughly ˆ (10.12)This proves the upper bound.
For the lower bound, observe that by the blowing up technique of Theorem 8.1 together with Theorem 3.1, there exist balls Since the (repeated) pre-images of the B * i are disjoint, and each repeated pre-image is a near-affine copy of B * i , the laminated area of i B * i is comparable to Thus, the lower bounds match the upper bounds up to a multiplicative constant.This concludes the proof of Theorem 1.2.

Limiting Vector Fields
In this section, we study the convergence of Blaschke products to vector fields.For a Blaschke product f a (z) = z d−1 i=1 z+a i 1+a i z , set z i := −a i .By a radial degeneration , we mean a sequence of Blaschke products f a ∈ B d such that: (1) The multiplier of the attracting fixed point tends (asymptotically) radially to e(p/q), i.e. arg(e(p/q) − a) → arg(e(p/q)).( 2) Each z i converges to some point e(θ i ) ∈ S 1 .
(3) The limiting ratios of speeds at which the zeros escape are well-defined, i.e.
To a radial degeneration, one can associate a natural measure µ on the unit circle which takes the escape rates into account: µ gives mass ρ i /q to e(θ i + j/q).Here, we use the convention that if some of the points coincide, we sum the masses.
For applications, it is convenient to use the convergence of linearizing coordinates: Corollary.As a → e(p/q) radially, the linearizing coordinates ϕ a : D → C converge to the linearizing coordinate ϕ κ := lim η→1 − g η (z)/η of (the semigroup generated by) the limiting vector field κ.
Remark.More generally, one can consider linear degenerations where a → e(p/q) asymptotically along a linear ray, i.e. with a ≈ e(p/q)(1 − δ + δ • T i) and δ small.In this case, the limiting vector field takes the more general form We call µ the driving measure and T the rotational factor.ζ−z dµ ζ is tangent to the unit circle, has simple poles and in between any two poles has a unique zero.Since a q → 1 radially, κ (0) = −1 which implies that 0 is a sink.Conversely, it can be shown that any vector field which satisfies the above properties comes from a radial degeneration of Blaschke products.Since we will not need this fact, we omit the proof.
where the estimate is uniform for a in any non-tangential sector at A.
Proof.This is an exercise in differentiation.One simply needs to compute and use the fact that 1 − |a| ≈ δ.
We first prove Theorem 11.1 in the case when a → 1.For a Blaschke product The idea is to compare f a (z) to the vector field κ(f a ) given by (11.3) with driving measure µ(f a ) = Lemma 11.2.We have the estimate: uniformly in the closed unit disk away from supp µ.
Proof.Using that (1 Theorem 11.1 now follows in the case when a → 1 since for radial degenerations, the rotational factor T (f a ) → 0. 11.2.Radial degenerations with a → e(p/q).As noted above, for a radial degeneration with a → e(p/q), we consider the limiting vector field of f •q a rather than of f a .In view of Lemma 11.2 above, to show that f •q a converges to a vector field κ whose driving measure gives mass ρ i /q to each point e(θ i + j/q), it suffices to analyze the zero set of f •q a .
Let us first consider the case of a generic radial degeneration, i.e. when the points e(θ i + j/q) with 1 ≤ i ≤ d − 1 and 0 ≤ j ≤ q − 1 are all different.The zero set of f •q a consists of the zeros of f a and their 1, 2, . . ., (q − 1)-fold pre-images.We omit the trivial zero at the origin and split the remaining zeros of f •q a into two groups: the dominant zeros and subordinate zeros.The dominant zeros are the zeros z i = z i,0 of f a (z) and their shadows z i,j near e(−jp/q)z i .We will refer to all other zeros as the subordinate zeros.From formula (7.3), it follows that the heights of the subordinate zeros are insignificant compared to the heights of the dominant zeros.Thus, only the dominant zeros contribute to the limiting vector field.
Let us now consider the general case.For a point z ∈ D, call w a dominant pre-image of z under f a if it is located near e(−p/q)ẑ, i.e. if | ŵ − e(−p/q)ẑ| ≤ .Otherwise, we say that w is a subordinate pre-image.We define a dominant zero of f •q a to be a point z ∈ D which is the j-fold dominant pre-image of some z i , with 0 ≤ j ≤ q − 1.To show that the driving measure µ has the desired expression, it suffices to show that the subordinate zeros have negligible height.We prove this in two lemmas: Multiplying over i = 1, 2, . . ., d − 1, we see that 11.3.Asymptotic semigroups.By an asymptotic semigroup on a domain Ω, we mean a family of holomorphic maps {f t } t≥0 : Ω t → C, with Ω t → Ω in the Carathéodory topology, satisfying for some holomorphic vector field κ.The notation O K denotes that the implicit constant is uniform on compact subsets of Ω.The condition (11.7) implies that We now show that the short term iteration of f t approximates the flow of κ : Theorem 11.3.Given a ball B(z 0 , R) compactly contained in Ω, one can find a t 0 > 0, so that for z ∈ B(z 0 , R) and t < t 0 , the limit over all possible partitions t 1 + t 2 + • • • + t n = t exists, and defines a holomorphic function on B(z 0 , R).
Above, the notation f P (z) denotes the expression f tn (f where P is a partition of the interval [0, t] by the points τ k = j≤k t j .The existence of the limit in (11.9) implies that {g t } satisfies g s • g t = g s+t as long as g s+t is well-defined.
Clearly, the vector field κ is the generator of the semigroup {g t }.
. Therefore, for 0 < t < t 0 , a finite partition P of the interval [0, t] defines a holomorphic function f P (z) on B(z 0 , R).
To prove the convergence of (11.9), it suffices to show that if Q is a refinement of P, then |f P (z)−f Q (z)| ≤ Ct||P||.Actually, it suffices to show that for an arbitrary partition P of [0, t], one has |f P (z) − f t (z)| ≤ Ct 2 .For this purpose, we introduce some bookkeeping: in view of (11.8), we say that the cost of splitting an interval of length T into two intervals is C • T 2 .Using the greedy algorithm, its not hard to show that the minimal cost of any partition of [0, t] is at most O(t 2 ).
To combine the "costs," we use the fact that on B(z 0 , R 1 ), the hyperbolic metric ρ B(z 0 ,R 2 ) is comparable to the Euclidean metric.Therefore, by the Schwarz lemma, which gives the claim.
Clearly, Theorem 11.2 is a special case of Theorem 11.3, where Ω = C \ P (κ) is the complement of the set of poles of κ.By the Schwarz lemma, inside the unit disk, g t (z) can be defined for all time, whereas on the unit circle, one can only define g t (z) until one hits a pole of κ.

Asymptotics of the Weil-Petersson metric
In this section, we show Theorem 1.3 which says that as a → e(p/q) radially in B 2 , the ratio ω B /ρ 1/4 → C q .It suffices to show that for a fixed λ with |λ| = 1, As noted in the introduction, the key to this result is the convergence of Blaschke products to vector fields.From the convergence of the linearizing coordinates (the corollary to Theorem 11.2), it follows that: Theorem 12.1.As a → e(p/q) radially, (i) The flowers F p/q (f a ) → F p/q (κ q ) in the Hausdorff topology.
(ii) The optimal Beltrami coefficients µ Together with Lemma 10.1, which controls the shapes of flowers near the unit circle, Theorem 12.1 implies the quasi-geodesic property: Lemma 12.1 (Quasi-geodesic property).As a → e(p/q) radially, each petal P ξ i (fa) (f a ) lies within a bounded distance of the geodesic ray [0, ξ i (f a )].Alternatively, the flower F(f a ) lies within a bounded neighbourhood of the hyperbolic convex hull of the origin and the ends ξ i (f a ).
For convenience of the reader, we give an alternative proof of the strong linearization property in Lemma 10.1 using the existence of the limiting vector field κ = κ q .Observe that as a → e(p/q) radially, the p/q-cycle ξ 1 (f a ), ξ 2 (f a ), . . ., ξ q (f a ) converges to the set of sources ξ 1 , ξ 2 , . . ., ξ q of κ.Also note that κ (ξ i ) is a positive real number since κ is tangent to the unit circle.
Choose a ball B(ξ i , R ) on which κ (z) κ (ξ i ) − 1 < 1/10.From the uniform convergence where C 1 = max z∈B(ξ i ,R ) |κ (z)| + .Therefore, for a sufficiently close to e(p/q), we have Flower counting hypothesis.From Theorem 10.2, we know that the immediate pre-flower is approximately the reflection of the flower about the critical point, while the pre-flowers are nearly-affine copies of the immediate pre-flower.Therefore, the convergence of flowers implies that the pre-flowers of all f a must also have nearly the same shape up to affine scaling.
Let n(r, f a ) denote the number of repeated pre-images of −a that lie in B(0, r).By renewal theory, for r close to 1, the circle S r intersects pre-flowers at "hyperbolically random" locations.Therefore, is reasonable to hypothesize that if µ = µ λ is an optimal Beltrami coefficient with |λ| = 1, then ||µ • χ G || 2 WP is proportional to c(f a ).To justify the intuition, we must show three things: * The contributions of the pre-flowers are more or less independent.
* All pre-flowers of the same size contribute roughly the same amount.* Most of the integral 2 3π ´|z|=1−r |v /ρ 2 | 2 dθ comes from pre-flowers whose size is comparable to 1 − r.
12.1.Decay of correlations.In this section, we use "flower" to mean either a flower or a pre-flower.Write the half-optimal coefficient as µ half = F µ F with µ F supported on F. Set Then v (z) = F v F (z).We wish to show that the integral average in (1.4) is proportional to the flower count.The main difficulty is that (1.4) features the L 2 norm so we have "correlations" 2 3π ρ 2 dθ.We now show that these correlations are insignificant compared to the main term 2 3π F ´|z|=r v F ρ 2 2 dθ.
For a point z ∈ D, let F z be the flower which is closest to z in the hyperbolic metric (in case of a tie, we pick F z arbitrarily) and R z be the union of all the other flowers.The integral average (1.4) splits into four parts:  Fz,Rz)   which is bounded by e −d D (0,a) ∼ (1 − |a|).The estimate for the other two error terms is similar.
12.2.Convergence of Beltrami coefficients.For a Blaschke product f a ∈ B 2 with a ≈ e(p/q), we define an idealized garden G id (f a ): The idealized flower F id (f a ) := F(g η ) is simply the flower of the limiting vector field.The idealized immediate preflower F id * (f a ) is the Möbius involution of F(g η ) about c(f a ).For a pre-flower F z (f a ), the idealized version F id z (f a ) is an affine copy of F id * (f a ), centered at z.We can define an idealized half-optimal Beltrami coefficient µ id in a similar manner: on F id (f a ), we let µ id • χ F id be the half-optimal Beltrami coefficient for the limiting vector field; while on the pre-flowers, we define µ id • χ F id z by scaling µ id • χ F id appropriately.For a Beltrami coefficient µ supported on the exterior unit disk, let us write: H[µ] := lim There are two sources of error: First, the pre-flowers don't quite match up with their idealized counterparts.Secondly, since the linearizing maps ϕ a and ϕ κ are slightly different, the Beltrami coefficients µ half and µ id themselves are slightly different.To prove Lemma 12.2, we define the "core-bulk-ends" decomposition of a flower.Namely, we split F α into three parts: the core C α δ , the bulk B α δ and the ends E α δ = i E α,i δ : We define the core, bulk and ends of a pre-flower F z as the pre-image of the corresponding part of F.
Estimating the symmetric difference.For a sufficiently close to e(p/q), the symmetric difference of the flower F and its idealized version F id is contained in  Estimating the difference between Beltrami Coefficients.From the convergence of the linearizing maps ϕ a → ϕ κ , it follows that when a is sufficiently close to e(p/q), the L ∞ norm of |µ half − µ id | is arbitrarily small on B 1/2+ (g η ).By Koebe's distortion theorem, an analogous statement holds for pre-flowers.Therefore, the difference |µ half • χ S − µ id • χ S | is small in L ∞ sense.Theorem 2.1 implies . This completes the proof of Lemma 12.2.
12.3.Flowers: large and small.We now show that most of the integral average 2 3π ´Sr |v /ρ 2 | 2 dθ comes from flowers whose size is (1 − r).In view of (12.1), for any > 0, we can find an 0 < r mix < 1 such that n(r, f a ) (1 − r) ≈ c(f a ), for r ∈ (r mix , 1).(12.5) For r ∈ (r mix , 1), write: From the lower bound, it follows that the integral 2 3π ´|z|=r |v med /ρ 2 | 2 dθ over only the medium flowers is c(f a ).We claim that if the "tolerance" k > 1 is large, then 12.4.An alternate route.To conclude, we give a slightly different approach to Theorem 1.3 by using renewal theory to estimate the integral average (1.4).The key is to show that most of the integral average comes from near the garden.Therefore, the part of the integral average over S r ∩ Ê2 is also small.The proof is complete.

µFigure 1 .
Figure 1.The support of the Beltrami coefficient takes up half of the quotient torus.

Figure 2 .
Figure 2. Gardens G(f a ) for the Blaschke products with a = 0.5 and 0.8.

Figure 3 .
Figure 3.A blow-up of G(f 0.5 ) near the boundary.A circle {z : |z| = r} with r close to 1 meets G(f 0.5 ) in small density.

Figure 5 .
Figure 5.The Mandelbrot set Equivalent definitions of the Weil-Petersson metric.Suppose f ∈ B d and f t , t ∈ (− , ) is a smooth path with f 0 = f , representing a tangent vector in T f B d .Consider the vector field v(z) := d dt t=0 H 0,t (z) where H 0,t : D → Ω − (f 0,t ) is the conformal conjugacy between f 0 and f 0,t .If f is a Blaschke product other than z → z d , one can define || ḟt || 2WP by the integral average (1.4), while if f (z) = z d , one can use a more complicated integral average described in[McM2]. below).
Proof.Part (i) follows from Lemma 6.1 as m c→0 (0, 1) = (−c, 1).To pin down the size and location of the immediate pre-petal, we use the fact that for a degree 2 Blaschke product, c is the hyperbolic midpoint of [0, −a].This implies that in the critically-centered picture, the center of the petal is m c→0 (0) = −c while the center of the immediate pre-petal is m c→0 (−a) = c.Therefore, by Koebe's distortion theorem, P−1 ⊂ B −1, const • 1 − |a| .Part (ii) follows by applying m 0→c to the last statement.
can work with the quantity n * (r, f a ) which counts the number of pre-flowers that intersect the circle S r = {z : |z| = r}.From the existence of the limiting shape and the quasi-geodesic property, it is easy to see that as a → e(p/q) radially, the quantities c(f a ) = lim r→1 n(r,fa) 1−r and c * (f a ) = lim r→1 n * (r,fa)

z) 2 2 dθ
By the lower bounds established in Section 10, the first term is bounded below by the flower count which decays roughly like 1 − |a|, while each of three other terms contribute on the order of O(1 − |a|), and so are negligible.Take for instance the second term.By the triangle inequality, for any z ∈ D,v Fz (z) ρ(z) 2 • v Rz (z) ρ(z) 2 e −d D (z,Fz) • e −d D (z,Rz) ≤ e −d D ( |v µ id /ρ 2 | 2 dθ − lim r→1 ˆ|z|=r |v µ half /ρ 2 | 2 dθ = o 1 − |a|as a → e(p/q).
The mating B d × B d → Rat d varies holomorphically with parameters.A natural way to put a complex structure on B d is via the Bers embedding B d → P d which takes a Blaschke product and mates it with z d to obtain a polynomial of degree d.Here the space P d ∼ = C d−1 is considered modulo affine conjugacy.The image of the Bers embedding is the generalized main cardioid in P d .
Question.For d ≥ 3, what is the completion of B d 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 B d ?
t) ∩ H and ζ ∈ B = {w : d H (z, w) < R}.Applications to Blaschke products.For a Blaschke product f ∈ B d , set δ c := min c (1 − |c|) where c ranges over the critical points of f that lie inside the unit disk.
1 (10.13)lying in the sectors S i for which ffl B i |v /ρ 2 (z)| 2 • |dz| 2 1.The reflection B * i of B i is a ball of definite hyperbolic size whose Euclidean center is located at height roughly