schwarzian versus a family of moving parabolic

SCHWARZIAN VERSUS A FAMILY OF MOVING PARABOLIC

POINTS

HANS HENRIK RUGH, LEI TAN, AND FEI YANG

Abstract. In this paper the trajectories of meromorphic quadratic differen-tials whose coefficients are Schwarzian derivatives of rational maps are studied.

We conjecture that the graphs consisting of the so-called negative-real critical

Schwarzian trajectories are finite for all rational maps of degree at least two.For any given rational map f , we construct a family of parabolic rational

maps {gw}w having a parabolic fixed point at the non-critical point w of f .

If w is a zero of the Schwarzian Sf , we prove that the orientations of the

negative-real critical trajectories of Sf (z)dz2 landing at w coincide with the

attracting directions of gw at w.The critical curve in every immediate parabolic basin of w is an arc starting

at w along the attracting direction and ending at a critical point of f , which

is a pole of Sf . We conduct some numerical experiments to compare thecritical curves in the immediate parabolic basins with the negative-real critical

Schwarzian trajectories, which support our conjecture.

1. Introduction

Let U ⊂ C be a domain. For a non-constant meromorphic function f : U → C,its Schwarzian derivative Sf is the meromorphic function defined by

Sf (z) =

f ′′′(z)

f ′(z)− 3

2

(f ′′(z)

f ′(z)

)2

if f(z) 6=∞,

limw→z

Sf (w) if f(z) =∞.

In particular, if f is a rational map, the quadratic differential Sf (z) dz2 is mero-morphic on the Riemann sphere, with poles located precisely at the critical pointsof f . Given a meromorphic function φ(z) on U , an inverse procedure is to find a

meromorphic function f : U → C such that Sf (z) = φ(z). In [CGRT16], somenecessary and sufficient conditions for a meromorphic quadratic differential withprescribed poles to be the Schwarzian derivative of a rational map were studied.

According to Thurston [Thu10], for a given rational map f , the meromorphicquadratic differential Sf (z) dz2 can be geometrically indicated by two perpendic-ular foliations (or trajectories): one set of streamlines where the quadratic formtakes positive real values (i.e., horizontal trajectories), and a perpendicular set oflines where it takes negative real values (i.e., vertical trajectories). Whenever thequadratic form has a simple zero, the foliations have singularities where the linesmake a Y pattern which branches in 3 different directions. At any critical point off , there is a double pole of the Schwarzian, where the positive real foliation circlesaround and the negative real foliation is radial (see [Str84, §7] or Theorem 2.1, andFigure 1). These lines show how the extension of f bends surfaces asymptoticallynear the complex plane.

Date: September 7, 2021.2020 Mathematics Subject Classification. Primary 30D30; Secondary 37F10, 37F75.Key words and phrases. Schwarzian derivatives; trajectories; parabolic rational maps.

1

2 HANS H. RUGH, LEI TAN, AND FEI YANG

Figure 1: A local view of the horizontal and vertical trajectories of a meromorphic

quadratic differential Sf (z)dz2 for a rational map f . The horizontal trajectories aredrawn by red dashed lines while the vertical trajectories are drawn by black solidlines. These two types of trajectories are perpendicular to each other. A simplezero and a double pole of the Schwarzian Sf (z) are marked by a hollow circle anda solid red dot respectively.

In other words, the geometric form of the data to recover a non-dynamical (i.e.,independent of the iteration) rational map, up to postcomposition with Mobiustransformations, is a quadratic differential having cylinders of equal circumferencenear the branch points, where the positive-real foliations consist of circumferencesof equal length. The negative-real leaves give a kind of current that matches thevertices and perpendicular to the positive-real ones. Therefore, to understand therational maps, it is helpful to study the foliations of the meromorphic quadraticdifferentials whose coefficients are their Schwarzian derivatives.

For a given rational map f , the horizontal (resp. vertical) trajectories of themeromorphic quadratic differential Sf (z)dz2 are also termed as positive-real (resp.negative-real) Schwarzian trajectories. They are idential to the (undirected) tra-

jectories of the vector field 1/√Sf (z) (resp. 1/

√−Sf (z)) (see §2). A trajectory

is called critical if its one end is a zero of Sf (z)dz2. To understand the foliationstructure of the quadratic differential Sf (z)dz2, it is useful to know the finitenessof all critical trajectories first.

For some cubic rational maps, the computer experiments show that the flow ofthe vector field dz/dt = 1/

√Sf (z) has recurrent trajectories, whereas that of the

vector field dz/dt = 1/√−Sf (z) has a finite critical graph. For the latter, it means

that every critical trajectory starting at a zero of Sf either reaches a pole of Sf(critical point of f) or makes a saddle connection (i.e., reaches another zero of Sf ).

In this paper, we are mainly interested in the following question:

Question. Is the graph consisting of all the critical trajectories (starting at thezeros) of the negative-real Schwarzian finite?

To study this question, we adopt an idea due to Buff and Epstein: viewing theproblem as a dynamical system. We construct a family of parabolic germs andstudy how the attracting directions of the parabolic petals change. The parabolicgerm at a non-critical point w of f is defined as gw := M−1

w ◦ f , where Mw is the

unique Mobius map that best approximates f at w, i.e., M(k)w (w) = f (k)(w) for

k = 0, 1, 2. If f is a rational map, then {gw : w ∈ C \ Crit(f)} is really a family

SCHWARZIAN VERSUS A FAMILY OF MOVING PARABOLIC POINTS 3

of rational maps of the same degree with a “moving” parabolic fixed point, whereCrit(f) is the set of all critical points of f (see §3).

We first establish a connection between the structure of the trajectories and thedynamical properties of the parabolic map gw.

Theorem 1.1. Let f be a rational map of degree d ≥ 2 and let Sf be the Schwarzianderivative of f . If f has exactly δ ≤ 2d − 2 distinct critical points, then Sf (z)dz2

has 2δ − 4 zeros in C (counted with multiplicity). Moreover, we have the followingtable regarding the classification of some special points of f and Sf , the trajectoriesof Sf (z)dz2 and the properties of gw:

f Sf posi-real traj. nega-real traj. gw =M−1w ◦ f

crit. point double pole circles radials —

non-crit. pt.non-zero regular regular double parabolic

order k zero k+2-branch star k+2-branch star k+2-parabolic

We will show that if w is a regular point of Sf , then the attracting (resp. re-

pelling) directions of the parabolic petals at w are exactly ± 1/√−Sf (w) (resp.

± 1/√Sf (w)) (see Lemma 3.2 and Corollary 3.3), which coincide with the direc-

tions of the negative-real (resp. positive-real) Schwarzian trajectories at w (see(2.3)). For the directions of the negative-real Schwarzian trajectories at the zerosof Sf , we have the following result.

Theorem 1.2. Let f be a rational map of degree d ≥ 2 and w be a zero of Sf .Then the negative-real Schwarzian trajectories at w are tangent to the attractingdirections of gw at w. Moreover, if d = 3, then w is a simple zero and there areprecisely 3 negative-real Schwarzian trajectories landing at w.

According to Fatou (see [Mil06, §10]), in every immediate parabolic basin of gwattaching at w, there exists a canonical critical curve (see (5.2)) connecting theparabolic fixed point w with a critical point of f (which corresponds to a doublepole of Sf ). We show (numerically) that in the case of non-saddle connection, suchcritical curves in parabolic basins of gw starting at the parabolic fixed point wapproximate the critical trajectories of the negative-real Schwarzian very well.

Note that each critical curve in the parabolic basin ends at one critical point of f .This leads us to think that the negative-real Schwarzian trajectories starting at thezeros may behave in the same way. This probably does not help in understandingthe Schwarzian, but does give information about the dynamics of the family {gw}w.Although we still cannot answer the question above, we hope our partial resultsand numerical experiments can shed some light on this problem.

This paper is organized as following: In §2, we calculate the directions of thenegative-real Schwarzian trajectories and prove Theorem 1.1 except the statementson the parabolic map gw. In §3, we define a family of parabolic rational maps,

{gw}w = {M−1w ◦ f}w, which is parameterized by w ∈ C \ Crit(f), prove Theorem

1.2 and the rest of Theorem 1.1. In §4, we study more properties of {gw}w and

focus mainly on the global structure of {gw : w ∈ C r Crit(f)}. In §5, we conductnumerical experiments on a family of specific cubic rational maps, and compare thecritical curves in immediate parabolic basins to critical trajectories of the negative-real Schwarzian. We also discuss the case of saddle connection and use the numericalexperiments to indicate that this case happens if and only if the 4 critical points ofthe cubic rational maps are concyclic.


Acknowledgements. This paper was written based on a group discussion led byBill Thurston in 2010, and the motivation can be found in Bill’s post on Math-overflow [Thu11]. We thank John H. Hubbard, Sarah Koch, Kevin Pilgrim andReinhard Schaefke for helpful discussions.

We are very grateful to the referee for his/her quick feedback, including encour-agements and valuable suggestions on the first version of this paper. We also thankthe referee for very careful reading, very helpful suggestions about the writing andorganization on the second version of this paper. F. Y. was supported by the NSFC(No. 12071210) and NSF of Jiangsu Province (No. BK20191246).

The preparation of this paper was sadly overshadowed by the early death of BillThurston (in August 2012) and Tan Lei (in April 2016). We dedicate this work totheir memory.

2. Quadratic differentials and trajectories

We first survey some preliminaries on quadratic differentials and trajectories,which can be found in [Str84]. Let S be a Riemann surface with a given complexstructure {Uα, hα}α∈Λ. A (meromorphic) quadratic differential ϕ on S is a set ofmeromorphic function elements ϕα in local parameters zα = hα(P ) for which thefollowing transformation law

ϕα(zα) dz2α = ϕβ(zβ) dz2

β , where dzβ =(dzβ

dzα

)dzα

holds whenever zα = hα(P ) and zβ = hβ(P ) are parameter values which correspondto the same point P ∈ Uα ∩ Uβ ⊂ S. The function element ϕα is also called therepresentation of the quadratic differential ϕ in terms of the parameter zα.

In our setting, S is the Riemann sphere C with the standard complex structure

{(C, id), (C r {0}, 1/z)}. Let z1 and z2 be the parameters corresponding to (C, id)

and (C r {0}, 1/z), respectively. For a given rational map f of degree at least two,one can verify that its Schwarzian

{Sf (z1) dz21 , Sf(1/z)(z2) dz2

2} (2.1)

is a meromorphic quadratic differential on C by using the following compositionformula

Sg◦f (z) = Sg(f(z)) · (f ′(z))2 + Sf (z). (2.2)

In the following, for simplicity we use Sf (z) dz2 to denote the quadratic differential(2.1) arising from the Schwarizian Sf .

It is known that the order of a zero or a pole of a quadratic differential ϕ isinvariant under a local coordinate transformation. The zeros and poles of ϕ arecalled the critical points and the others are called regular points of ϕ.

Definition (Trajectories). In a neighborhood of every regular point of ϕ(z) dz2,

in terms of the natural representation using the local parameter w =∫ √

ϕ(z) dz,the quadratic differential takes the form dw2. A smooth curve γ along

arg dw2 = argϕ(z) dz2 = θ = const., where θ ∈ [0, 2π),

is called a straight arc (with respect to the quadratic differential ϕ).A horizontal trajectory of the quadratic differential ϕ is a maximal horizontal arc

for which θ = 0 and a vertical trajectory is a maximal vertical arc for which θ = π.Thus, under this natural representation, a horizontal (resp. vertical) trajectory inthe z-plane corresponds to a horizontal (resp. vertical) line in the w-plane.


Let γ = γ(t) with t ∈ (0, 1) be a continuous parametrization of a trajectory γ ofa quadratic differential ϕ(z) dz2. Suppose that the following two limits exist (γ isseen in the z-plane):

z0 = limt→0+

γ(t) ∈ C and θ = limt→0+

arg(γ(t)− z0

)∈ [0, 2π).

Then we say that γ lands at z0, and use ν = eiθ to denote the orientation of γ(with respect to the z-plane and z0).

A trajectory of ϕ(z) dz2 is called critical if one of its ends is a zero of ϕ(z) dz2.Sometimes we say that the critical trajectory starts at the zero for convenience.A critical trajectory is called a saddle connection if both of its ends are zeros ofϕ(z) dz2. The local structures of the critical trajectories of quadratic differentialsare characterized by the following theorem.

Theorem 2.1 ([Str84, § 7]). Let ϕ(z) dz2 be a quadratic differential defined on aRiemann surface.

(a) Suppose that P is a zero of ϕ of order k ≥ 1. Then in the natural repre-

sentation w =∫ √

ϕ(z) dz, there are exactly k + 2 horizontal (and k + 2vertical) trajectories landing at w(P ) whose angle in a circular segment hasbeen subdivided into k + 2 equal sectors.

(b) Suppose that P is a pole of ϕ of order 2 and ϕ has the representation(aζ−2) dζ2 in some distinguished parameter ζ. If a ∈ R and a < 0, thenthe horizontal (resp. vertical) trajectories near P in terms of ζ are circlesaround the origin (resp. are radials starting at the origin).

For a given rational map f of degree at least two, any maximal smooth arcsatisfying Sf (z) dz2 > 0 (resp. Sf (z) dz2 < 0) is a horizontal (resp. vertical)trajectory of Sf (z) dz2. On the other hand, note that the trajectories of the vectorfields

dz

dt=

1√Sf (z)

anddz

dt=

1√−Sf (z)

, (2.3)

respectively, are exactly the horizontal and vertical trajectories of Sf (z) dz2. Thesetwo kinds of trajectories are called positive-real (resp. negative-real) Schwarziantrajectories. In the following we often omit the word “real” for simplicity.

Proposition 2.2. Let w be a zero of Sf (z) dz2 of order k ≥ 1. Then there areexactly k+2 negative Schwarzian trajectories landing at w whose orientations {νj :

1 ≤ j ≤ k + 2} satisfy νk+2j = −e−iθ, where θ = argS

(k)f (w).

Proof. According to Theorem 2.1(a), there are exactly k + 2 negative Schwarziantrajectories γ1, · · · , γk+2 landing at w. Without loss of generality, we may assumethat w = 0 and that in a neighborhood of the origin one has:

Sf (z) = akzk +O(zk+1), where k ≥ 1 and ak 6= 0.

By (2.3), we have

zk(

dz

dt

)2

= − 1

ak(1 +O(z)),

which is equivalent to

zk+2 + higher order terms = −(k + 2

2

)21

akt2.

This implies that the orientations ν1, · · · , νk+2 of the trajectories γ1, · · · , γk+2 at

w, respectively, satisfy νk+2j = −e−iθ, where θ = arg ak = argS

(k)f (0). �


Proof of Theorem 1.1 except the statement on gw. Let f be a rational map of de-gree d ≥ 2. If w is not a critical point of f , then w is not a pole of Sf . If w ∈ C isa critical point of f , i.e., degw(f) = k ≥ 2, then one can write f as

f(z) = bk(z − w)k +O((z − w)k+1), as z → w,

where bk 6= 0. From a direct calculation we have

Sf (z) =1− k2

2(z − w)2

(1 +O

(z − w

))as z → w. (2.4)

Hence w is pole of Sf and the pole of the Schwarzian Sf is always a double pole.Since 1 − k2 < 0 for all k ≥ 2, by Theorem 2.1(b), the positive (resp. negative)Schwarzian trajectories near the pole w are all circles (resp. radials).

Suppose w is not a critical point of f . If further w is an order k zero of Sf withk ≥ 1, then there are exactly k + 2 horizontal (and k + 2 vertical) trajectories ofSf (z)dz2 landing at w whose angle in a circular segment has been subdivided intok+ 2 equal sectors by Proposition 2.2. If w is not a zero of Sf , then w is a regularpoint of Sf (z)dz2.

Finally, let δ ≥ 1 be the number of distinct critical points of f . Since everypole of Sf (z) dz2 is a double pole, it implies that Sf (z) dz2 has exactly 2δ polescounted with multiplicity. Let g(·) denote the genus of a compact Riemann surface.It follows from the Riemann-Roch Theorem (see [FK92, p. 76]) that

]Zero(Sf (z) dz2)− 2δ = 4g(C)− 4 = −4.

Therefore, Sf (z) dz2 has 2δ − 4 zeros. In particular, if the critical points of f areall simple, then Sf (z) dz2 has 4d− 8 zeros. �

3. A family of moving parabolic points

In geometric terms, the Schwarzian derivative measures the infinitesimal changein cross ratio caused by the meromorphic function f (see [Sie10]). In particular, Sfis identically zero precisely for Mobius transformations.

3.1. Mobius transformations and the parabolic family. Let f be a rationalmap of degree d ≥ 2. We use Crit(f) to denote the collection of all the critical

points of f . For each w ∈ CrCrit(f), let Mw be the unique Mobius transformationsatisfying

Mw(w) = f(w), M ′w(w) = f ′(w) and M ′′w(w) = f ′′(w). (3.1)

Note that if w =∞ or f(w) =∞, we adopt the generalized definition of derivative(i.e., the spherical derivative using coordinate transformation). A direct calculationshows that

Mw(z) = f(w) +f ′(w)(z − w)

1− f ′′(w)2f ′(w) (z − w)

and

M−1w (z) = w +

z − f(w)

f ′(w) + f ′′(w)2f ′(w) (z − f(w))

. (3.2)

One can show that the determinant of the coefficients of Mw is 4(f ′(w))3, i.e., Mw

exists if and only if w is not a critical point of f .

Denote

gw(z) = M−1w ◦ f(z). (3.3)


Then gw is a rational map and deg(gw) = deg(M−1w ) · deg(f) = deg(f). By (3.1),

both f(z) and Mw(z) have the following expansion as z → w:

f(w) + f ′(w)(z − w) +f ′′(w)

2(z − w)2 +O

((z − w)3

).

Therefore, we have gw(w) = w, g′w(w) = 1 and w is a parabolic fixed point of gwwith multiplier 1. In fact, suppose gw(z) = w+a1(z−w)+a2(z−w)2 +O

((z−w)3

)as z → w. By expanding the both sides of Mw ◦ gw(z) = f(z) at w and comparingthe coefficients, we have a1 = 1 and a2 = 0. Hence

gw(z) = z +O((z − w)3

)as z → w. (3.4)

Remark. As a special case1, suppose f(z) = z + O(z3) is a polynomial. Thenfor w = 0 we have Mw(z) = z and gw(z) = f(z). This is not the case for f(z) =z + a2z

2 +O(z3) with a2 6= 0.

Sometimes we may encounter the case w = ∞ and this is not convenient forcalculations. We show that one can reduce the case w =∞ to w ∈ C.

Lemma 3.1. If the rational map f is replaced by f = f ◦M , where M is a Mobius

transformation, then the corresponding gw with w ∈ C rCrit(f) defined in (3.3) isreplaced by gw = M−1 ◦ gw ◦M , where M(w) = w. So the dynamics of the familiesgw and gw are conjugate by a Mobius transformation.

Proof. Let w and w be two given points in C such that w = M(w) and w is not acritical point of f . By the definition of Mw, we have

M (k)w (w) = f (k)(w), where k = 0, 1, 2.

Then we have

(Mw ◦M)(k)(w) = (f ◦M)(k)(w) = (f)(k)(w), where k = 0, 1, 2.

Note that Mw ◦M is a Mobius transformation which is the best approximation of

f at w. By the uniqueness of Mw with regard to f , we have Mw = Mw ◦M and

gw = M−1w ◦ f = M−1 ◦M−1

w ◦ f ◦M = M−1 ◦ gw ◦M.

The proof is finished. �

Since SM−1w

is identically zero, we have Sf = Sgw by (2.2) and (3.3). This impliesthat we can understand Sf in terms of studying gw.

Lemma 3.2. Let g(z) = z+az3 +O(z4) be a holomorphic map defined in a neigh-borhood of 0. Then Sg(0) = 6a and if a 6= 0, the map g has a double parabolic point

at 0 whose two attracting directions are ± 1/√−a/|a| and two repelling directions

are ± 1/√a/|a|.

Proof. As g′(0) = 1, g′′(0) = 0 and g′′′(0) = 6a, we see that Sg(0) = 6a. Ifg′′′(0) 6= 0, the map g has a double parabolic fixed point at 0. The two attractingdirections are ± ν so that a

|a|ν2 = −1 (see [Mil06, §10, p. 104]), or equivalently

ν = ± 1/√−a/|a|. Therefore the two attracting directions are ± 1/

√−a/|a| and

two repelling directions are ± 1/√a/|a|. �

Corollary 3.3. In a small neighborhood of w ∈ C that is not a critical point of f ,the map gw has expansion

gw(z) = z +Sf (w)

6(z − w)3 +O((z − w)4).

1We would like to thank the referee for providing such an interesting example.


Proof. Let Mw be as in (3.1). By (3.4) there exists a ∈ C such that gw canbe written as gw(z) = z + a(z − w)3 + O((z − w)4). By Lemma 3.2, we havea = Sgw(w)/6 = Sf (w)/6. �

Remark. (1) One can observe that if f is “closer” to a linear map near w, thenSf (w) is smaller and gw is “closer” to the identity.

(2) If w =∞, one can obtain a similar expansion of gw as in Corollary 3.3 by acoordinate transformation. See Lemma 3.1.

Definition (p-parabolic). The germ g(z) = z + azp+1 + O(zp+2) with a 6= 0 istangent to the identity of order (or multiplicity) p + 1 ≥ 2 at the origin. Forsimplicity, in this paper g is called p-parabolic and 0 is called a p-parabolic fixed

point of g. A p-parabolic fixed point z ∈ C is defined similarly.

It follows from Lemma 3.2 and Corollary 3.3 that at a regular point w of Sf(recall that w is regular if Sf (w) 6= 0 or ∞), the map gw has a double parabolic

fixed point at w whose attracting directions are parallel to ± 1/√−Sf (w). At

the zeros of the Schwarzian Sf , it follows from Corollary 3.3 that the parabolicmultiplicity increases. In particular we have the following precise result:

Lemma 3.4. Let w ∈ C be an order k ≥ 0 zero of Sf . Then w is a (k+2)-parabolicfixed point of gw.

Proof. The statement for k = 0 follows from Corollary 3.3. Without loss of gener-ality, we assume that w = 0 is a zero of Sf of order k ≥ 1. Then in a neighborhoodof w = 0, the map gw = g0 can be written as

g0(z) = z + azp+1 +O(zp+2) with a 6= 0 and p ≥ 3,

and Sg0(z) has the following form:

Sg0(z) =g′′′0 (z)

g′0(z)− 3

2

(g′′0 (z)

g′0(z)

)2

= a(p+ 1)p(p− 1)zp−2 +O(zp−1).

Since Sf (z) = Sg0(z) and 0 is an order k ≥ 0 zero of Sf , it follows that p − 2 = kand hence 0 is a (k + 2)-parabolic fixed point of g0. �

Proof of Theorem 1.1 for the statement on gw. This is an immediate consequenceby combining Corollary 3.3 and Lemma 3.4. �

Remark. If f has a p-parabolic fixed point at w ∈ C with p ≥ 2, then f(w) = w,f ′(w) = 1 and f ′′(w) = 0. The Mobius transformation Mw satisfying (3.1) mustbe the identity. Thus gw = M−1

w ◦ f = f . This implies that any rational maphaving a p-parabolic fixed point with p ≥ 2 can be realized by some gw, and thusthe dynamics of gw can be arbitrarily complicated.

Definition (Petals, see [Mil06, p. 111]). Let z ∈ C be a p-parabolic fixed pointof g which is defined and univalent in some neighborhood N of z and let νj bean attracting direction at z, where 1 ≤ j ≤ p. An open set P ⊂ N is called anattracting petal of g for the direction νj at z, if

• g maps P into itself; and• an orbit z0 7→ z1 7→ · · · under g is eventually absorbed by P if and only if

it converges to z from the direction νj .

The open set P ⊂ N is called a repelling petal of g for the direction νj at z if P isan attracting petal of g−1 for the direction νj at z.


Theorem 3.5 (Leau-Fatou Flower Theorem, [Mil06, Theorem 10.7, p. 112]). If zis a p-parabolic fixed point (i.e., a fixed point of multiplicity p+ 1 ≥ 2), then withinany neighborhood of z there exist simply connected petals Pj, where the subscriptj ranges over the integers modulo 2p and where Pj is either repelling or attractingaccording to whether j is even or odd. Furthermore, these petals can be chosen sothat the union

{z} ∪ P0 ∪ · · · ∪ P2p−1

is an open neighborhood of z. When p > 1, each Pj intersects each of its twoimmediate neighbors in a simply connected region Pj ∩Pj±1 but is disjoint from theremaining Pk.

In the following, we use petal to denote the attracting petal for short unlessotherwise is specified. Note that for each attracting direction, the petal is far fromunique. For example, the image of any petal is also a petal. Hence in the following,two petals corresponding to the same attracting direction are seen to be the samepetal. If z is a p-parabolic fixed point of g, by Theorem 3.5 there are exactly p ≥ 1petals corresponding to p different attracting directions at z.

Lemma 3.6. Let g be a rational map of degree d ≥ 2. Suppose that z is a fixedpoint of g with g′(z) = 1. Then g has at most d petals at z.

Proof. Without loss of generality, we may assume that z = 0. If the origin has atleast d+ 1 petals, then g can be expanded in a small neighborhood of 0 as:

g(z) = z(1 + azd+k +O(zd+k+1)) with a 6= 0 and k ≥ 1.

This, however, would imply that g has at least d+ k+ 1 fixed points (counted withmultiplicity) in C by Rouche’s Theorem, which is a contradiction since g has at

most d+ 1 fixed points (counted with multiplicity) in C. �

Remark. In particular, if g is a polynomial of degree d satisfying g(z) = z andg′(z) = 1, then g has at most d− 1 petals at z.

Corollary 3.7. Let f be a cubic rational map and gw = M−1w ◦f be the map defined

in (3.3). Then

(a) gw has at least 2 and at most 3 petals at each w ∈ C r Crit(f); and(b) The zeros of Sf are all simple.

Proof. (a) Note that gw is defined when w is not a critical point of f (i.e., apole of Sf ). By Corollary 3.3 and Theorem 3.5, gw has at least 2 petals at each

w ∈ C rCrit(f). By Lemma 3.6, gw has at most 3 petals at each w ∈ C rCrit(f).

(b) Suppose w is a zero of Sf of order k ≥ 1. According to Lemma 3.4, w is a(k + 2)-parabolic fixed point of gw and hence gw has k + 2 petals at w. However,by Lemma 3.6, we have k + 2 ≤ 3. This implies that k = 1. In particular, w is asimple zero and gw has exactly 3 petals at w. �

We are in the position to prove the following result.

Theorem 1.2. Let f be a rational map of degree d ≥ 2 and w be a zero of Sf .Then the negative-real Schwarzian trajectories at w are tangent to the attractingdirections of gw at w. Moreover, if d = 3, then w is a simple zero and there areprecisely 3 negative-real Schwarzian trajectories landing at w.

Proof. Without loss of generality, we assume that w = 0 is a zero of Sf of orderk ≥ 1. According to the proof of Lemma 3.4, in a neighborhood of 0 we have

gw(z) = g0(z) = z + azk+3 +O(zk+4) where a 6= 0, and

Sf (z) = Sg0(z) = a(k + 1)(k + 2)(k + 3)zk +O(zk+1).


By [Mil06, p. 104], the direction of each attracting direction ι of gw (k + 2 in all)at the origin satisfies a

|a| ιk+2 = −1. On the other hand, by Proposition 2.2, the

orientation ν of each of the k + 2 negative Schwarzian trajectories landing at wsatisfies νk+2 = −e−iθ with θ = arg a. This means that the negative Schwarzianderivative trajectories at w = 0 and the attracting directions of gw at w have thesame directions. The statement for cubic rational maps follows immediately fromCorollary 3.7(b). �

3.2. The limit of gw at the critical points. Note that the Mobius transforma-tion Mw and the rational map gw are not defined at the critical points of f . Inthe rest of this section, we assume that the critical points of f are all simple, i.e.,]Crit(f) = 2d− 2. The following lemma shows that gw degenerates when w tendsto a critical point of f (i.e., tends to a pole of Sf ).

Lemma 3.8. Let w0 be a critical point of f . Then gw → w0 on C as w →w0. Moreover, the convergence is uniform outside of any open neighborhood off−1(f(w0)).

Proof. By Lemma 3.1, without loss of generality, we may assume that w0 6=∞ andf(w0) 6=∞. By (3.2) and (3.3), one can write gw as

gw(z) = w +

(f ′(w)

f(z)− f(w)+f ′′(w)

2f ′(w)

)−1

. (3.5)

By the assumption that the critical points of f are all simple, we have f ′(w0) = 0

and f ′′(w0) 6= 0. If z ∈ C is a point for which f(z) 6= f(w0), then gw(z) tends to

w0 in C r f−1(f(w0)) as w → w0. Moreover, gw tends to w0 uniformly outside ofany open neighborhood of f−1(f(w0)).

In a small neighborhood of w0, we have f(w) = f(w0) + f ′′(w0)2 (w − w0)2 +

O((w − w0)3). If z ∈ C is a point for which f(z) = f(w0), then

f ′(w)

f(z)− f(w)=f ′(w)− f ′(w0)

f(w0)− f(w)=

f ′′(w0)(w − w0) +O((w − w0)2)

− f′′(w0)

2 (w − w0)2 +O((w − w0)3)

= − 2

w − w0+O(1).

By (3.5) and note that f ′′(w0) 6= 0, it follows that gw(z) tends to w0 for z ∈f−1(f(w0)). Therefore, we have gw → w0 on C as w → w0. �

Remark. In general, the convergence gw → w0 in Lemma 3.8 is not uniform on

the whole C. For example, suppose that f(z) = z2 and w0 = 0. Then gw(z) =

w + 1/( 2wz2−w2 + 1

2w ) tends to 0 on C but not uniformly as w → 0.

Lemma 3.8 shows that as w tends to a critical point of f , then gw tends to the

critical point itself on C. We now consider the limit of gw up to a linear rescaling.

Lemma 3.9. Let w0 6= ∞ be a critical point of f . For each w 6= w0 and ∞, letNw be the unique Mobius transformation fixing ∞ that maps 0 to w and 1 to w0.

Then N−1w ◦ gw ◦Nw tends to a parabolic Mobius transformation on C as w → w0.

Proof. By Lemma 3.1, without loss of generality, we may assume that f(w0) 6=∞and f(w0) 6= f(∞). Note that Nw(z) = w + (w0 − w)z. By (3.5), we have

N−1w ◦ gw ◦Nw(z) =

2(f ′(w)− f ′(w0))

w0 − w·(

2(f ′(w))2

f(w + (w0 − w)z)− f(w)+ f ′′(w)

)−1

.


Note that

2(f ′(w))2

f(w + (w0 − w)z)− f(w)=

2(f ′(w)− f ′(w0))

(w0 − w)z· (w0 − w)z

f(w + (w0 − w)z)− f(w)· f ′(w).

If z 6= 0 and ∞, we have

limw→w0

N−1w ◦ gw ◦Nw(z) = −2f ′′(w0) ·

(−2f ′′(w0)

z+ f ′′(w0)

)−1

=z

1− z2

.

If z = 0, then N−1w ◦ gw ◦ Nw(0) = 0. If z = ∞, then f(z) 6= f(w0) by the

assumption. We obtain

limw→w0

N−1w ◦ gw ◦Nw(∞) = lim

w→w0

2(f ′(w)− f ′(w0))

w0 − w· 1

f ′′(w0)= −2.

Therefore, N−1w ◦ gw ◦Nw(z) tends to the parabolic Mobius map z 7→ z

1−z/2 on Cas w → w0. �

Remark. If w is sufficiently close to a critical point w0 of f (but w 6= w0), thenSf (w) 6= 0. By Corollary 3.3, the map N−1

w ◦gw ◦Nw has two petals at N−1w (w) = 0

if w is sufficiently close to w0. However, the limit z1−z/2 has only one.

4. The structure of the collection of gw

Let Ratd be the space of rational maps of degree d ≥ 2. We say that g ∈ Ratdis equivalent to f if g = M ◦ f , where M ∈ PSL(2,C) is a Mobius transformation.Let

[f ] := {g ∈ Ratd : g = M ◦ f, where M ∈ PSL(2,C)}.Then as a subset of Ratd, [f ] forms an equivalence class.

Lemma 4.1. Let f ∈ Ratd with d ≥ 2. Then

(a) For any f1, f2 ∈ [f ], we have Crit(f1) = Crit(f2) and Sf1 = Sf2 ; and(b) The map M 7→M ◦ f defined from PSL(2,C) to Ratd is injective.

Proof. (a) All the maps in [f ] have the same critical points and the same Schwarzianderivatives by (2.2).

(b) Suppose that M1 ◦ f = M2 ◦ f , where M1,M2 ∈ PSL(2,C). Then we havef = M−1

1 ◦M2 ◦ f . This implies that M−11 ◦M2 is the identity since it fixes at least

three different points in C. Then M1 = M2. �

By Lemma 4.1(b), the equivalence class [f ] is a 3-dimensional subset of Ratd.Let

〈f〉 := {g ∈ [f ] : g has a p-parabolic fixed point, where p ≥ 2}.For each w ∈ C r Crit(f), let gw = M−1

w ◦ f ∈ Ratd be the map defined in (3.3).By Corollary 3.3, gw has a p-parabolic fixed point at w with p ≥ 2. Therefore, wehave a map

Ψ : C r Crit(f)→ 〈f〉which is defined as Ψ(w) = gw.

Lemma 4.2. For each f ∈ Ratd with d ≥ 2, the map Ψ : C r Crit(f) → 〈f〉 is asurjection and it is at most [d+1

3 ] to one. In particular, Ψ is a bijection if d ≤ 4.

Proof. Every g = M−1 ◦ f ∈ 〈f〉 with M ∈ PSL(2,C) has a p-parabolic fixed point

at a point w ∈ C with p ≥ 2. Hence

M−1 ◦ f(w) = w, (M−1 ◦ f)′(w) = 1 and (M−1 ◦ f)′′(w) = 0.


By the first two equations above, we have f(w) = M(w) and (M−1)′(f(w))·f ′(w) =f ′(w)/M ′(w) = 1. This is equivalent to

M(w) = f(w) and M ′(w) = f ′(w). (4.1)

Note that

(M−1◦f)′′(z)|z=w =( f ′(z)

M ′(M−1 ◦ f(z))

)′∣∣∣z=w

=f ′′(w)M ′(w)− f ′(w)M ′′(w)

(M ′(w))2= 0.

Since w is not a critical point of f , by (4.1), we have M ′(w) = f ′(w) 6= 0. Thismeans that M ′′(w) = f ′′(w). Combining with (4.1), it follows that the Mobiustransformation M also satisfies (3.1). According to the uniqueness of Mw, it impliesthat M = Mw and hence Ψ(w) = g. Therefore, Ψ is surjective.

For any g ∈ 〈f〉 and any w ∈ Ψ−1(g), w is a p-parabolic fixed point of g withp ≥ 2. Hence g has p + 1 fixed points at w counted with multiplicity. Since g hasexactly d + 1 fixed points, this means that Ψ−1(g) is a finite set. Suppose thatΨ−1(g) = {w1, · · · , wk} and wi 6= wj if i 6= j, where k ≥ 1 and 1 ≤ i, j ≤ k. Then

wi is a pi-parabolic fixed point of g with pi ≥ 2 and we have∑ki=1(pi + 1) ≤ d+ 1.

Therefore, 3k ≤ d + 1 and Ψ is at most [d+13 ] to one. In particular, if d ≤ 4, then

Ψ is an injection and hence is a bijection. �

Example. There exists a family of rational maps of degree 5 such that Ψ is notinjective. Define

f(z) =z + a2z

2 + a3z3 + a4z

4 + a5z5

1 + a2z + b3z2 + a4z3 + a5z4, where a5 6= 0 and a3 6= b3.

One may verify that both 0 and ∞ are 2-parabolic fixed points of f . In particular,we have Mw = id for w = 0 or ∞. Therefore, Ψ(0) = M−1

0 ◦ f = M−1∞ ◦ f = Ψ(∞).

Let f ∈ Ratd with d ≥ 2 and w ∈ C r Crit(f). By Lemma 4.2, one can write〈f〉 as

〈f〉 = {M−1w ◦ f : w ∈ C r Crit(f)}. (4.2)

Proposition 4.3. Let f ∈ Ratd with d ≥ 2. If f1, f2 ∈ [f ], then 〈f1〉 = 〈f2〉.

Proof. Let fi = Mi ◦ f ∈ [f ], where i = 1, 2. For any f ∈ 〈f1〉, one can write

f = M ◦ f1, where M is a Mobius transformation and f has a p-parabolic fixed

point in C with p ≥ 2. Note that f can be also written as f = (M ◦M1 ◦M−12 )◦f2.

This means that f ∈ 〈f2〉 and hence 〈f1〉 ⊂ 〈f2〉. Similarly, we have 〈f2〉 ⊂ 〈f1〉.Therefore, 〈f1〉 = 〈f2〉. �

Hence 〈f〉 is a set depending on [f ], but not on the particular choice of f .

Proposition 4.4. For each f ∈ Ratd with d ≥ 2 and M ∈ PSL(2,C), we have

〈f ◦M〉 = {M−1 ◦ h ◦M : h ∈ 〈f〉}.

Proof. By (4.2), every h ∈ 〈f〉 can be written as h = M−1w ◦f , where w ∈ CrCrit(f).

Since h has a p-parabolic fixed point at w with p ≥ 2, it follows that M−1 ◦ h ◦M = (M−1 ◦M−1

w ) ◦ (f ◦M) has also a p-parabolic fixed point at M−1(w). ThenM−1 ◦ h ◦M ∈ 〈f ◦M〉. In particular, we have {M−1 ◦ h ◦M : h ∈ 〈f〉} ⊂ 〈f ◦M〉.

On the other hand, each g ∈ 〈f ◦M〉 can be written as g = N ◦ f ◦M for someN ∈ PSL(2,C) such that g has a p-parabolic fixed point at w with p ≥ 2. Sinceg = N ◦f ◦M = M−1◦(M ◦N ◦f)◦M , it follows that M ◦N ◦f has also a p-parabolicfixed point at M(w). Therefore, M ◦N ◦ f ∈ 〈f〉 and g ∈ {M−1 ◦ h ◦M : h ∈ 〈f〉}.This implies that 〈f ◦M〉 ⊂ {M−1 ◦ h ◦M : h ∈ 〈f〉}. �


5. Numerical experiments on cubic rational maps

For the vertical trajectories of Schwarzian Sf (z) dz2, we are interested in twotypes of points: the first is Crit(f), which consists of all the critical points of f(i.e., the poles of Sf ) and the second is Zero(Sf ), consisting of all the zeros of Sf .We know that at each point of Zero(Sf ), there are 3 or more directions along whichthe vertical trajectories land (see Theorem 2.1(a)). As stated in the introduction,we conjecture that any one of such critical trajectories ends at another critical pointof Sf (z) dz2, i.e., a point in either Crit(f) or Zero(f).

In this section, we conduct some numerical experiments to support our conjec-ture. In particular, we focus our attention on the following family of cubic rationalmaps:

fα(z) :=2α+ 3z2 + αz3

1 + 3αz + 2z3, where α ∈ C and α6 6= 1. (5.1)

A direct calculation shows that

f ′α(z) = −6(z3 − 1)(z − α2)

(1 + 3αz + 2z3)2.

Hence the critical points of fα are 1, ξ := e2πi/3, ξ2, α2 and accordingly, the criticalvalues are fα(1) = 1, fα(ξ) = ξ2, fα(ξ2) = ξ and

fα(α2) =2α+ 3α4 + α7

1 + 3α3 + 2α6=

2α+ α4

1 + 2α3.

This implies that 1 is a super-attracting fixed point, {ξ, ξ2} forms a super-attracting2-cycle, and α2 is a free critical point (i.e., the forward orbit of α2 is unconstrained).

Remark. The family fα was introduced in [BEKP09, §4], which was used to finda critically finite rational map such that the corresponding Thurston pullback mapis a ramified Galois covering and has a fixed critical point.

Given a set C of 2d − 2 distinct points in C, there are finitely many choicesof quadratic differentials ϕi(z) dz2 with double poles at C that are Schwarzianderivatives of degree d rational maps. A natural question is whether each ϕi(z) dz2

depends holomorphically on C. In [CGRT16, §6], the family fα was used to serveas an example to show that Sfα(z) dz2 does not depend holomorphically on C ={1, ξ, ξ2, α2}.

Note that for α6 6= 1, every fα has exactly 4 different critical points. ThenSfα(z) dz2 has exactly 4 double poles by (2.4), and 4 simple zeros by Theorem 1.1and Corollary 3.7(b).

Experiment 1 (see Figure 2). For the family fα, considering α = 0.45 andα = i respectively, we obtain two collections of critical trajectories of the nega-tive Schwarzian of fα. From the pictures of Sfα(z) dz2 in Figure 2, one can seethat the 8 critical points of Sfα(z) dz2 (i.e., 4 simple zeros and 4 double poles)induce 12 germs of edges (critical trajectories) and we have following two types ofcombinatorics:

(i) There are exactly 3 edges (i.e., vertical critical trajectories) starting at eachzero of Sfα(z) dz2 and each of them terminates at a pole of Sfα(z) dz2 (seethe left picture in Figure 2);

(ii) There are exactly two saddle connections (i.e., both end points of the verticalcritical trajectories are zeros of Sfα(z) dz2, colored blue) linking 4 germs ofedges, and the remaining connections are received by at least one pole (seethe right picture in Figure 2).


ξ2

ξ

1α2

ξ2

ξ

1α2

Figure 2: The vector fields and the critical trajectories of negative Schwarizianderivatives of cubic rational maps fα with α = 0.45 and i respectively. The polesand the zeros of Sfα are marked by hollow circles and solid dots respectively. Thegreen circles are the unit circle, which are used to locate the relative positions ofthe critical points 1, ξ, ξ2 and α2 of fα. The leftmost and rightmost horizontal linesin the left picture, which are connected by the infinity, belong to the same criticaltrajectory. As can be seen from the pictures, both critical graphs are finite. Thepictures are drawn by the open source software Scilab 5.5.2.

Theorem 5.1 (Fatou coordinate, [Mil06, §10]). Let g be a rational map having a

p-parabolic fixed point at z ∈ C, where deg(g) ≥ 2 and p ≥ 1. Then every immediateparabolic basin U of z contains at least one critical point of g and there exists aholomorphic map Φ : U → C satisfying Φ(g(z)) = Φ(z) + 1. The map Φ is calledthe Fatou coordinate of g|U and it is unique up to an additive constant.

Let gw = M−1w ◦f be the parabolic rational map defined in (3.3) based on a given

rational map f . Suppose that f is cubic, then gw is also cubic and it is defined for

all w ∈ C r Crit(f). If w0 is a zero of Sf , then gw0 has a 3-parabolic fixed pointat w0 by Corollaries 3.3 and 3.7. Hence w0 has 3 immediate parabolic basins, andeach of them contains at least one critical point by Theorem 5.1 (and at most oneof them contains two critical points since gw has exactly 4 critical points). Let Ube one of these three immediate parabolic basins.

Definition (Critical curve). If U contains exactly one critical point c of f , wenormalize the Fatou coordinate Φ of gw|U such that Φ(c) = 0 and call the curve

γ :=

{The closure of the connected component ofΦ−1((0,+∞)) which is invariant under gw

}(5.2)

the critical curve in U . Note that γ is an analytic curve connecting the criticalpoint c with the parabolic fixed point w0. See left three pictures in Figure 3.

If U contains two distinct critical points c and c′, we define the critical curveas following: we first normalize Φ such that Φ(c) = 0 and define γ as in (5.2). Ifc′ 6∈ γ, then we call γ a critical curve in U . If c′ ∈ γ, we then normalize Φ such thatΦ(c′) = 0 and define a new γ as in (5.2) and call this γ the critical curve. Notethat in this case, U may contain two critical curves. See Figure 5 for an example.

By Theorem 1.2, we know that each critical curve has the same (tangent) di-rection as one critical trajectory of the negative Schwarzian at the parabolic fixed

https://www.scilab.org/


ξ2

ξ

w1

α2 1

ξ2

ξ

w1

ξ2

ξ

α2

w2

1

ξ2

ξ

α2w2

ξ2

ξ

w3

α2 1

ξ

w3

α2 1

Figure 3: The pictures on the left show the Fatou components of gw, where w = wi(i = 1, 2, 3) are zeros of Sfα with α = 0.45 (we don’t include w = w4 since thepictures are symmetric to those of w3 about the real axis). The three immediateparabolic basins and their preimages are colored white, gray and dark gray re-spectively. The three critical curves in the parabolic basins are colored blue. Weinclude these pictures in the planes on the right which show the critical trajectoriesof the negative Schwarzian (see the picture in Figure 2 on the left). It seems verylikely that the three critical curves coincide with the corresponding three criticaltrajectories. The pictures on the left are drawn by writing C language programs.


point. If these two curves coincide, then one can conclude that the critical graph ofthe negative Schwarzian is finite (at least holds for a large number of cubic rationalmaps). The question is: Are they coincident?

Experiment 2 (see Figure 3). We still work on the cubic rational map fα. By adirect calculation, we have

Sfα(z) = −3(1 + 4α3 − (4α− 8α4)z − 18α2z2 + (8− 4α3)z3 + (4α+ α4)z4

)2 (α2 − z)2(1− z3)2

.

For α = 0.45, the 4 zeros of Sfα are

w1 ≈ −4.53706588, w2 ≈ −0.54628583, w3,4 ≈ 0.46794580± 0.28295330i.

Therefore, gw has a 3-parabolic fixed point at w = wi for 1 ≤ i ≤ 4. We draw thecritical curves in the parabolic basins of gw and then compare them with the criticaltrajectories of the negative Schwarzian via putting them in the same pictures (seealso Figure 2).

Experiment 3 (see Figure 4). From Experiment 2, although the critical curvesand the corresponding critical trajectories starting at wi are close, where 1 ≤ i ≤ 4,our sufficiently precise numerical experiments show that they are not coincident.See Figure 4 for a specific example for w2.

ξ

w2

Figure 4: A critical trajectory (black curve) starting at the zero w2 of Sfα(z)dz2

with α = 0.45, the forward critical orbit (red dots) of ξ = e2πi/3 under the parabolicmap gw2

and their successive zooms. These three pictures are zooms of the middlepicture in Figure 3 on the right. It seems that the critical trajectory ends at thecritical point ξ. The critical orbit is “almost” contained in the trajectory in theleftmost picture, but not quite as illustrated by the rightmost picture.


Experiment 4 (see Figures 5 and 6). In Experiment 1, we see that the criticaltrajectories of Sfα(z) dz2 have saddle connections for α = i (see Figure 2 on theright). Actually, our numerical experiments show that if α is contained in the unitcircle (but α6 6= 1), then the 4 critical points of fα are concyclic, and there are 2saddle (or homoclinic) connections, i.e., the trajectories go from a zero of Sfα toanother.

If there is a saddle connection, i.e., the trajectory go from a zero to another,then we cannot use the critical curve in the immediate parabolic basin of gw toapproximate the saddle connection since the critical curve connects the zero withthe pole of Schwarzian but not the zero with another zero.

Let fα be the map defined in (5.1) with α = i. By a direct calculation, one canverify that Sfα has exactly 4 zeros:

w′1 ≈ + 0.09164031 + 0.45727992i, w′2 = 1/w′1,

w′3 ≈ − 0.63114460− 0.29926278i, w′4 = 1/w′3.

Then gw has a 3-parabolic fixed point at w = w′i for all 1 ≤ i ≤ 4. We draw thetwo critical curves in one immediate parabolic basin of gw starting at w = w′1. SeeFigure 5.

These two critical curves cannot approximate the critical trajectory well. How-ever, if we choose the normalization of Fatou coordinate Φ in that basin such thatΦ(w′3) = 0, then one can see that the modified “critical curve” starting at w′1 andending at w′3 approximates the critical trajectory very well. See Figure 6.

ξ

ξ2

1α2

w′1

ξ

ξ2

1α2

w′1

Figure 5: The Fatou components of gw, where w = w′1 is a zero of Sfα with α = i.The two critical curves passing through two critical points respectively are coloredred. In this case they cannot approximate the critical trajectory of the negativeSchwarzian as there is a saddle connection. Compare with Figure 6.

Note that in the above 4 experiments, we only draw the corresponding picturesfor two values of α, i.e., 0.45 and i. In fact, these two values are not special at alland they are just two representatives. Using C language programs and the opensource software Scilab 5.5.2, we have made several animations by the computer, toexplore the relations between the critical trajectories of the negative Schwarziansof fα and the critical curves in the immediate parabolic basins of gw. Indeed, thenumerical experiments show that such two types of curves lie very close, as shownin Figures 3 and 6.

https://www.scilab.org/


ξ

ξ2

1α2

w′1

ξ

ξ2

w′1

w′3

α2 1

Figure 6: The picture on the left shows the Fatou components of gw, where w = w′1a zero of Sfα with α = i. The horizontal trajectory under the Fatou coordinate thatpasses through another zero w′3 is colored red. We bring this picture to the planeon the right which shows the critical trajectories of the negative Schwarzian (seethe picture in Figure 2 on the right). It seems very likely that this curve coincideswith the corresponding critical trajectory of the negative Schwarzian.

Unfortunately, up to now we can only prove that these two types of curves havethe same starting orientations (see Proposition 2.2), but cannot give an mathemat-ical explanation why they can lie so close. We think this is a subject worthy offurther study.

References

[BEKP09] X. Buff, A. L. Epstein, S. Koch, and K. Pilgrim, On Thurston’s pullback map, Complexdynamics, A K Peters, Wellesley, MA, 2009, pp. 561–583.

[CGRT16] G. Cui, Y. Gao, H. H. Rugh, and L. Tan, Rational maps as Schwarzian primitives,

Sci. China Math. 59 (2016), no. 7, 1267–1284.[FK92] H. M. Farkas and I. Kra, Riemann surfaces, second ed., Graduate Texts in Mathemat-

ics, vol. 71, Springer-Verlag, New York, 1992.

[Mil06] J. Milnor, Dynamics in one complex variable, third ed., Annals of Mathematics Stud-ies, vol. 160, Princeton University Press, Princeton, NJ, 2006.

[Sie10] P. Siegel, Is there an underlying explanation for the magical powers of the Schwarzian

derivative?, Mathoverflow, Question: 38105, 2010.[Str84] K. Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete

(3), vol. 5, Springer-Verlag, Berlin, 1984.

[Thu10] W. Thurston, An anwer to “Is there an underlying explanation for the magical powersof the Schwarzian derivative?”, Mathoverflow, Question: 38105, 2010.

[Thu11] , What are the shapes of rational functions, Mathoverflow, Question: 38274,2011.

Departement de Mathematiques, Universite Paris-Saclay, Orsay 91405, France

E-mail address: [email protected]

Faculte des sciences, LAREMA, Universite d’Angers, Angers 49045, France

Department of Mathematics, Nanjing University, Nanjing 210093, P. R. ChinaE-mail address: [email protected]

https://mathoverflow.net/questions/38105

https://mathoverflow.net/questions/38105

http://mathoverflow.net/questions/38274

schwarzian versus a family of moving parabolic

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