dynamics, topology and bifurcations of mcmullen maps

Appl. Math. J. Chinese Univ.2013, 28(4): 494-504

Dynamics, topology and bifurcations of McMullen maps

WANG Xiao-guang YIN Yong-cheng

Dedicated to Professor Kien-Kwong Chen on the Occasion of his 120th Birthday

Abstract. We give a survey of many different topological structure arise in the dynamical and

parameter planes of McMullen maps.

§1 Introduction

The complex dynamical systems came from the study of Newton’s method to find roots ofreal-valued functions. In 1879, Arthur Cayley [3] first noticed the difficulties in generalizingthe Newton’s method to complex roots of polynomials with degree greater than 2 and complexinitial values. This opened the way to the study of the theory of iterations of holomorphicfunctions, initiated by Pierre Fatou and Gaston Julia around 1920. A very good introductionto this field is Milnor’s book [25].

Let C be the complex plane, and C be the Riemann sphere. The Riemann sphere C isa topological sphere S2 endowed a complex structure. A non-constant holomorphic self mapf : C → C can be written as a rational map in local coordinates:

f(z) =adz

d + · · · + a0

bdzd + · · · + b0,

where d is a positive integer and |ad|+ |bd| �= 0. For each point w ∈ C, the number of preimagesin f−1(w), counting multiplicity, is exactly d, which is called the degree (or topological degree)of f . For d ≥ 1, let Ratd be the space of rational maps of degree d. When d = 1, Rat1 consistsof all the Mobius transformations

z �→ az + b

cz + d

with ad− bc �= 0. The space Rat1 carries a group structure.Let f ∈ Ratd for some d ≥ 1. We are interested in the behavior of the forward orbits

f(z), f2(z) = f(f(z)), f3(z) = f(f(f(z))), · · · of the points z ∈ C. When d = 1, this is wellknown based on the classification of the Mobius transformations. So in the following discussion,we assume d ≥ 2.

The behavior of the forward orbits of z under the map f enables us to decompose theRiemann sphere C into two subsets: the Fatou set and the Julia set. The Fatou set F (f)

Received: 2013-10-15.MR Subject Classification: 37F45, 37F10, 37F15.Keywords: Fatou set and Julia set, McMullen map, hyperbolic component, Jordan curve.Digital Object Identifier(DOI): 10.1007/s11766-013-3224-5.

WANG Xiao-guang, YIN Yong-cheng. Dynamics, topology and bifurcations of McMullen maps 495

consists of the stable points. A point z ∈ C is called stable if there is a neighborhood U ofz such that the maps {fn : U → C;n ≥ 1} is a normal family. One may easily verify thatthe Fatou set is an open set. The Julia set J(f) is the complement of the Fatou set. Thesetwo sets are totally invariant: f−1(J(f)) = J(f) and f−1(F (f)) = F (f). They also satisfy:F (fn) = F (f) and J(fn) = J(f).

The multiplier of a point z of period p is the derivative (fp)′(z) of the first return map. Themultiplier provides a first approximation to the local dynamics of fp. Accordingly, we say z is

repelling if |(fp)′(z)| > 1;indifferent if |(fp)′(z)| = 1;attracting if |(fp)′(z)| < 1; andsuperattracting if (fp)′(z) = 0.An indifferent point is parabolic if (fp)′(z) is a root of unity.Here are some basic properties of the Julia sets:

Theorem 1.1. Let f be a rational map of degree at least two. Then its Julia set J(f) satisfies:1. J(f) �= ∅ and it is perfect: it has no isolated points.2. Repelling periodic points are dense in J(f).3. The inverse orbit of every point in J(f) is dense in J(f).4. If an open set U meets J(f), then J(f) ⊂ fn(U) for some n ≥ 1.

A component of Fatou set is called a Fatou component. A fundamental problem, going backto Fatou and Julia, is that whether there exist wandering Fatou components (the componentsU such that fm(U) �= fn(U) for any m > n ≥ 0). This problem is resolved by Dennis Sullivan[38] in 1985:

Theorem 1.2 (Sullivan). Every Fatou component U is pre-periodic. That is, there are twointegers n ≥ 0, p ≥ 1 such that fn(U) = fn+p(U).

§2 McMullen maps

The rational maps on the Riemann sphere C = C ∪ {∞}

fλ : z �→ zd +λ

zm, λ ∈ C

∗ = C \ {0}, d,m ≥ 1

regarded as a singular perturbation of the monomial z �→ zd, take a simple form but exhibitvery rich dynamical behavior. These maps are known as ‘McMullen maps’, since McMullen [21]first studied these maps and pointed out that when (d,m) = (2, 3) and λ is small, the Julia setis a Cantor set of circles. This family attracts many people for several reasons. The notableone is probably that the Julia set varies in several classic fractals. It can be homeomorphic toeither a Cantor set, or a Cantor set of circles, or a Sierpinski carpet [10]. Another reason isthat this family provides many examples for different purpose to understand the dynamic ofrational maps. We refer the reader to [4, 5, 6, 9, 10, 11, 15, 29, 31, 38] and the reference thereinfor a number of related results. See also Devaney’s survey [7] on the study of these maps.

The critical set of fλ is {0,∞} ∪ Cλ, where Cλ consists of d + m critical points εcλ withεd+m = 1 and cλ is one critical point of fλ. The Julia set J(fλ) has (d + m)-fold symmetry:e2πi/(d+m)J(fλ) = J(fλ). Let Bλ be the Fatou component containing ∞ and Tλ the componentof f−1

λ (Bλ) containing 0. It is possible that Bλ = Tλ.

496 Appl. Math. J. Chinese Univ. Vol. 28, No. 4

For any k ≥ 0, we define a parameter set Hk as follows:

Hk = {λ ∈ C∗; k is the first integer such that fk

λ(Cλ) ⊂ Bλ}.

A component of Hk is called a escape domain of level k. One may verify that H0 = {λ ∈C

∗; fλ(Cλ) ⊂ Bλ},H1 = ∅ and Hk = {λ ∈ C∗; fk−1λ (Cλ) ⊂ Tλ �= Bλ} for k ≥ 2. The

complement of the escape domains is called the non-escape locus M.

Figure 1: The Julia sets in the family fλ : z �→ z3 + λz3 : a Cantor set (upper-left), a Cantor set

of circles (upper-right), a Sierpinski curve (lower-left) and a connected set (lower-right).

Theorem 2.1 (Escape Trichotomy [10] and Connectivity [12]).1. If λ ∈ H0, then J(fλ) is a Cantor set.2. If λ ∈ H2, then J(fλ) is a Cantor set of circles.3. If λ ∈ Hk for some k ≥ 3, then J(fλ) is a Sierpinski curve.4. If λ ∈ M, then the Julia set J(fλ) is connected.

On the 25th Anniversary of the Mandelbrot set in the Snowbird Conference, Bob Devaney[9] posed the following problems:

Problem 2.2 (Devaney). Consider the family fλ : z �→ zn + λzn with n ≥ 3.

1. In the dynamical plane, is ∂Bλ always a Jordan curve when the Julia set J(fλ) isconnected?

2. In the parameter plane, are the boundaries of escape domains Jordan curve? It’s knownin [5] that ∂H2 is.

Answering these problems involves the Branner-Hubbard-Yoccoz puzzle theory, to be intro-duced in the next section.


§3 The Branner-Hubbard-Yoccoz puzzle theory

The Branner-Hubbard-Yoccoz puzzle theory is a powerful tool in complex dynamical sys-tems. It first appeared in Branner-Hubbard’s work on cubic polynomials [1] and Yoccoz’s workon quadratic polynomials [16]. See also Milnor’s lecture notes [26]. Further applications includethe study of rigidity of polynomials [17, 27], Branner-Hubbard conjecture [18, 30], topology ofFatou components [35] and even topology of hyperbolic components in the parameter plane[32, 33], etc. The aim of this section is to give a brief introduction to this theory.

Let X,X ′ be connected open subsets of C with finitely many smooth boundary componentsand such that X ′ ⊂ X �= C. A holomorphic map f : X ′ → X is called a rational-like map(resp. a simple rational-like map) if it is proper and has finitely many critical points (resp. asingle critical point with multiplicity one) in X ′. We denote by deg(f) the topological degreeof f and by K(f) =

⋂

n≥0 f−n(X) the filled Julia set, by J(f) = ∂K(f) the Julia set.

Although we do not use here, we remark that an analogue of Douady-Hubbard’s straighten-ing theorem [13] also holds for rational-like maps. That is, the rational-like map f : X ′ → X ishybrid equivalent to a rational map R of degree deg(f); if K(f) is connected, such R is uniqueup to Mobius conjugation if we require that it is postcritically finite outside its filled Julia set(c.f. [40], Theorem 7.1).

A finite, connected graph Γ is called a puzzle if it satisfies the conditions: ∂X ⊂ Γ, f(Γ ∩X ′) ⊂ Γ, and the orbit of each critical point avoids Γ.

The puzzle pieces Pn of depth n are the connected components of f−n(X \ Γ) and the onecontaining the point x is denoted by Pn(x). For any x ∈ J(f), let orb(x) = {x, f(x), f2(x), · · · }be the forward orbit of x. For n ≥ 0, let P ∗

n(x) = Pn(x) if orb(x)∩Γ = ∅, and P ∗n(x) =

⋃

x∈PnPn

if orb(x) ∩ Γ �= ∅. The impression Imp(x) of x is defined by

Imp(x) =⋂

P ∗n(x).

A puzzle Γ is said to be k-periodic at a critical point c if fk(Pn+k(c)) = Pn(c) for anysufficiently large n. A puzzle Γ is said admissible if for each critical point c ∈ J(f), there is aninteger dc > 0 such that Pdc(c) \ Pdc+1(c) is a non-degenerate annulus.

Theorem 3.1 (Branner-Hubbard [1], Roesch [34], Yoccoz [26, 16]). Let f : X ′ → X be a simplerational-like map, with critical point c ∈ K(f). Suppose that Γ is an admissible puzzle.

1. If Γ is not periodic at c, then K(f) = J(f) and for any x ∈ J(f), the impressionImp(x) = {x}.

2. If Γ is k-periodic at c, then fk : Pn+k(c) → Pn(c) for some large n defines a quadratic-likemap with filled Julia set Imp(c). Moreover,

Imp(x) =

{

a conformal copy of Imp(c), if x ∈ ⋃

k≥0 f−k(Imp(c)),

{x}, if x ∈ K(f) − ⋃

k≥0 f−k(Imp(c)).

Here is the idea of the proof: Let Ad(x) = P ∗d (x) \P ∗

d+1(x). When Γ is not periodic at c, bysome delicate combinatorial analysis, we can find a sequence of integers {nk, k ≥ 1} such that{Ank

(c), k ≥ 1} is a sequence of non-degenerate annuli whose moduli have complex bounds (i.e.a uniform lower bound). Thus

+∞∑

k=1

mod(Ank(c)) = +∞.


This implies, by Grotzsch inequality, that the impression Imp(c) = {c}. Thus for any x ∈ J(f)whose forward orbit avoids the graph Γ, the conformal property of f transfers the moduli ofthe non-degenerate critical annuli to that of An′

k(x), k ≥ 1. This yields Imp(x) = {x}. For the

points x ∈ J(f) whose forward orbits meet the graph Γ, the backward contraction of f alongthe orbits implies that Imp(x) = {x}. When Γ is k-periodic at c, the idea is same.

In general, Theorem 3.1 is used to study the topology (e.g. connectivity and local connec-tivity) of the Julia sets. By applying complex analysis especially some distortion results, onecan further study some analytic properties (e.g. Lebesgue measure and Hausdorff dimension)of the Julia sets. One of the fundamental results is due to Lyubich:

Theorem 3.2 (Lyubich). Let f : X ′ → X be a simple rational-like map, with critical pointc ∈ K(f). Suppose that Γ is an admissible puzzle. Then

1. If Γ is not periodic at c, then the Lebesgue measure of J(f) is zero.2. If Γ is periodic at c, then Lebesgue measure of J(f) is zero if and only if the Lebesgue

measure of ∂Imp(c) is zero.

Theorem 3.2 is slightly stronger than Lyubich’s original result [19], but the proof goesthrough without any problem.

§4 Topology in the dynamical plane

The answer of Devaney’s first problem is

Theorem 4.1 ([29]). For any n ≥ 3 and any λ ∈ C∗,

• ∂Bλ is either a Cantor set or a Jordan curve. In the latter case, all Fatou componentseventually mapped to Bλ are Jordan domains.

• If ∂Bλ is a Jordan curve containing neither a parabolic point nor the recurrent critical setCλ, then ∂Bλ is a quasi-circle.

Here, the critical set Cλ is called recurrent if Cλ ⊂ J(fλ) and the set ∪k≥1fkλ (Cλ) has an

accumulation point in Cλ. The proof of Theorem 4.1 is based on the Branner-Hubbard-Yoccozpuzzle theory. To apply this theory, we need to construct a kind of Jordan curve which cuts theJulia set into two connected parts. These curves are called cut rays. They play a crucial rolein the study of both the dynamical plane and the parameter plane. For this, we briefly sketchtheir constructions here.

To begin, we identify the unit circle S = R/Z with (0, 1]. We define a map τ : S → S

by τ(θ) = nθ mod 1. Let Θk = ( k2n ,

k+12n ] for 0 ≤ k ≤ n and Θ−k = ( k

2n + 12 ,

k+12n + 1

2 ] for1 ≤ k ≤ n− 1. Obviously, (0, 1] = ∪−n<j≤nΘj .

Let Θ be the set of all angles θ ∈ (0, 1] whose orbits remain in⋃n−1

k=1 (Θk ∪ Θ−k) under alliterations of τ . One may verify that Θ is a Cantor set. Given an angle θ ∈ Θ, the itinerary ofθ is a sequence of symbols (s0, s1, s2, · · · ) ∈ {±1, · · · ,±(n− 1)}N such that τk(θ) ∈ Θsk

for allk ≥ 0. The angle θ ∈ Θ and its itinerary (s0, s1, s2, · · · ) satisfy the identity ([29], Lemma 3.1):

θ =12

(

χ(s0)n

+∑

k≥1

|sk|nk+1

)

,

where χ(s0) = s0 if 0 ≤ s0 ≤ n and χ(s0) = n− s0 if −(n− 1) ≤ s0 ≤ −1.


Note that eπi/(n−1)fλ(z) = (−1)nfe2πi/(n−1)λ(eπi/(n−1)z) for all λ ∈ C∗. This implies that

the fundamental domain of the parameter plane is

F0 = {λ ∈ C∗; 0 ≤ argλ < 2π/(n− 1)}.

We denote the interior of F0 by

F := {λ ∈ C∗; 0 < argλ < 2π/(n− 1)}.

In our discussion, we assume λ ∈ F0 and let Oλ = ∪k≥0f−kλ (∞) be the grand orbit of ∞.

Let c0 = c0(λ) = 2n√λ be the critical point that lies on R+ := [0,+∞) when λ ∈ R+ and varies

analytically as λ ranges over F . Let ck(λ) = c0ekπi/n for 1 ≤ k ≤ 2n − 1. The critical points

ck with k even are mapped to v+λ while the critical points ck with k odd are mapped to v−λ .

Let k = ck[0,+∞] be the closed straight line connecting 0 to ∞ and passing through ckfor 0 ≤ k ≤ 2n− 1. The closed sector bounded by k and k+1 is denoted by Sλ

k for 0 ≤ k ≤ n.Define Sλ

−k = −Sλk for 1 ≤ k ≤ n− 1. These sectors are arranged counterclockwise about the

origin as Sλ0 , S

λ1 , · · · , Sλ

n , Sλ−1, · · · , Sλ

−(n−1).The critical value v+

λ always lies in Sλ0 because arg c0 ≤ arg v+

λ < arg c1 for all λ ∈ F0.Correspondingly, the critical value v−λ lies in Sλ

n . The image of k under fλ is a straightray connecting one of the critical values to ∞; this ray is called a critical value ray. As aconsequence, fλ maps the interior of each of the sectors of Sλ±1, · · · , Sλ

±(n−1) univalently onto a

region Υλ, which can be identified as the complex sphere C minus two critical value rays.

Theorem 4.2 (Cut ray, [6] [29]). For any λ ∈ F and any angle θ ∈ Θ with itinerary(s0, s1, s2, · · · ), the set

Ωθλ :=

⋂

k≥0

f−kλ (Sλ

sk∪ Sλ

−sk)

is a Jordan curve intersecting the Julia set J(fλ) in a Cantor set.

Theorem 4.2 is originally proven for the parameters λ ∈ F ∩M in [29]. The proof actuallyworks for all λ ∈ F without any difference.

Here are some facts about the cut rays: Ωθλ = −Ωθ

λ and Ωθλ = Ωθ+1/2

λ ; Rλ(θ) ∪Rλ(θ+ 12 ) ⊂

Ωθλ∩F (fλ) ⊂ ∪k≥0f

−kλ (Bλ); 0,∞ ∈ Ωθ

λ and Ωθλ\{0,∞} is contained in the interior of Sλ

s0∪Sλ−s0

;fλ(Ωθ

λ) = Ωτ(θ)λ and fλ : Ωθ

λ → Ωτ(θ)λ is a two-to-one map. We refer the reader to [29] for more

details of the cut rays.After the construction of cut rays, the second crucial step in Branner-Hubbard-Yoccoz puzzle

theory is to find an admissible graph. By an elaborate choice, we can show (see [29], Proposition4.5):

Proposition 4.3. For any n ≥ 3 and any λ with λ ∈ F , if fλ is not post-critically finite, thenthere always exists an admissible graph Γ.

Now we sketch the strategy of the proof. By a result of Devaney, to show that ∂Bλ is aJordan curve when it is connected, it suffices to show that it is locally connected. When λ isa positive and real number, it is easy to show that there are at most two post-critical pointson ∂Bλ. The local connectivity follows from Tan-Yin’s Theorem [39] (in the case that fλ has aparabolic cycle) or reduced to Theorem 4.4, which implies the local connectivity of ∂Bλ(in thecase that fλ has no parabolic cycle).


0Rλ(1)Rλ(1/2)

Figure 2: An example of cut ray: Ω1λ = Ω1/2

λ . (n = 3)

The main issue is to deal with the case λ ∈ F ∩ M. In this case, we apply the Branner-Hubbard-Yoccoz theory. By Theorem 3.1 and Proposition 4.3, we see that:

When Γ is not periodic at any c ∈ Cλ, then Imp(x) = {x} for any x ∈ ∂Bλ. So ∂Bλ islocally connected.

When Γ is periodic at some c ∈ Cλ, we can show that the post-critical set P (fλ) intersects∂Bλ at only finitely many points (the proof of this fact is very technical, see [29], Proposition7.3). We may exclude the geometrically finite case which is easy to handle. It turns out thatfλ satisfies the backward contraction property on ∂Bλ. We summarize the result for this caseas follows:

Theorem 4.4 (Backward contraction on ∂Bλ,[29]). Suppose that λ ∈ C∗\H0 and ∂Bλ containsneither a parabolic point nor the recurrent critical set Cλ, then fλ satisfies the following propertyon ∂Bλ: there exist three constants δ0 > 0, C > 0 and 0 < ρ < 1 such that for any 0 < δ < δ0,any z ∈ ∂Bλ, any integer k ≥ 0 and any component Uk(z) of f−k

λ (B(z, δ)) that intersects with∂Bλ, Uk(z) is simply connected with Euclidean diameter diam(Uk(z)) ≤ Cδρk.

As a consequence of Theorem 4.4, if λ ∈ ∂H0, then ∂Bλ contains either the critical set Cλ

or a parabolic point of fλ.Once we get Theorem 4.4, we can show that ∂Bλ is locally connected (see [29], Section 6).

This suffices for our purpose.

§5 Topology in the parameter plane

This section comes from the paper [28].In the parameter space, people are interested in the distribution of hyperbolic maps, topology

of hyperbolic components, etc. Recall that a rational map is hyperbolic if all critical orbits areattracted by the attracting cycles, see [25, 24]. The hyperbolic maps form an open set in theparameter space. Each component of this open set is called a hyperbolic component.


Let’s still look at the following family of McMullen maps:

fλ : z �→ zn +λ

zn, n ≥ 3.

A McMullen map fλ is hyperbolic if the free critical orbit orb(cλ) is attracted by either ∞ oran attracting cycle in C. The topology of hyperbolic components of this family is studied byDevaney, Roesch and Steinmatz [5][31][38]. It turns out that these components have simpletopology. To state the theorems, we first recall some definitions and notations:

For any k ≥ 0, we define a parameter set Hk as follows:

Hk = {λ ∈ C∗; k is the first integer such that fk

λ(Cλ) ⊂ Bλ}.

A component of Hk is called an escape domain of level k. The complement of the escapedomains is called the non-escape locus M. It can be written as

M = {λ ∈ C∗; fk

λ (Cλ) does not tend to infinity as k → ∞}.

The set M is invariant under the maps z �→ z and z �→ e2πin−1 z.

The Bottcher map φλ of fλ is defined in a neighborhood of ∞ by φλ(z) = limk→∞

(fkλ(z))n−k

. It

is unique if we require φ′λ(∞) = 1. The map φλ satisfies φλ(fλ(z)) = φλ(z)n and φλ(eπi/nz) =eπi/nφλ(z). One may verify that near infinity,

φλ(z) =∑

k≥0

ak(λ)z1−2kn, a0(λ) = 1, a1(λ) = λ/n, · · · .

If λ ∈ C∗ \ H0, then both Bλ and Tλ are simply connected. In that case, there is a uniqueRiemann mapping ψλ : Tλ → D, such that ψλ(w)−n = φλ(fλ(w)) for w ∈ Tλ and ψ′

λ(0) = n√λ.

The free citical values of fλ are denoted by v±λ = ±2√λ.

Theorem 5.1 (Parameterization of escape domains[5][31][38]).1. H0 is the unbounded component of C∗ − M. The map Φ0 : H0 → C − D defined by

Φ0(λ) = φλ(v+λ )2 is a conformal isomorphism.

2. H2 is the component of C∗ − M containing the punctured neighborhood of 0. Theholomorphic map Φ2 : H2 → C − D defined via Φ2(λ)n−2 = φλ(fλ(v+

λ ))2 and limλ→0 λΦ2(λ) =2

2n2−n , is a conformal isomorphism .

3. Let H be a escape domain of level k ≥ 3. The map ΦH : H → D defined by ΦH(λ) =ψλ(fk−2

λ (v+λ )) is a conformal isomorphism.

We know from Theorem 5.1 that every hyperbolic component is isomorphic to either theunit disk D or D∗ = D−{0}. Devaney’s second problem concerns the topology of the boundaryof these hyperbolic components. The answer is

Theorem 5.2. The boundaries of all hyperbolic components are Jordan curves.

The topological structure of the boundary of the unbounded hyperbolic component H0

(consisting of the parameters for which the Julia set J(fλ) is a Cantor set) is also interesting.We show:

Theorem 5.3. Cusps are dense in ∂H0.


Figure 3: Parameter plane of McMullen maps, n = 3, 4.

Here, according to McMullen [23], a parameter λ ∈ C∗ is called a cusp if the map fλ

has a parabolic cycle on ∂Bλ. Geometrically, a cusp is a point where the bifurcation locusis cusp-shaped. A theorem of McMullen says that cusps are dense on the Bers’ boundary ofTeichmuller space [22]. The analogue of McMullen’s Theorem in the world of rational map, asa conjecture posed by McMullen [23], would be the density of geometrically finite parabolicson the boundary of the hyperbolic component Ud of degree d polynomials containing zd. Thisconjecture is verified by Roesch [32] for the one-dimensional slice zd + czd−1 (d ≥ 3 and c is aparameter) and remains open in full generality. The result we prove here is, in the sprit, relatedto this conjecture. But it has to be thought of a phenomenon in some one-dimensional slices ofrational maps.

Let’s sketch how to obtain Theorem 5.3. Assuming Theorem 5.2, one gets a canonicalparameterization ν : S → ∂H0, where ν(θ) is defined to be the landing point of the parameterray R0(θ) (defined as Φ−1

0 ((1,+∞)e2πiθ)) in H0. We actually give a complete characterizationof ∂H0 and its cusps:

Theorem 5.4 (Characterization of ∂H0 and cusps).1. λ ∈ ∂H0 if and only if ∂Bλ contains either Cλ or a parabolic cycle.2. ν(θ) is a cusp if and only if npθ ≡ θ mod Z for some p ≥ 1.

Theorem 5.3 is an immediate consequence of Theorem 5.4 since {θ;npθ ≡ θ mod Z, p ≥ 1}is a dense subset of the unit circle S.

Let’s sketch the idea of the proof of Theorem 5.2, whose detail is carried out in [28]. Toprove that ∂H0 is a Jordan curve, we first use the dynamical result: Theorem 4.4. This yieldsthe following correspondence: If a parameter ray R0(θ) accumulates on a point λ, then whenwe look at the dynamical plane, the external ray Rλ(θ) lands at either the critical value v+

λ ora parabolic cycle. In either case, the Lebesgue measure of Julia set is zero. To show the localconnectivity of ∂H0, we show that any two maps (which are not cusps) in the same impressionof the parameter ray are quasiconformally conjugate. This conjugacy is constructed with thehelp of cut rays and it is holomorphic in the Fatou set. The ‘zero measure statement’ impliesthat the conjugacy is actually a Mobius map. So the two maps are the same. After we knowingthe local connectivity, a dynamical result (Theorem 4.1) enables us to show that the boundary


is a Jordan curve.The proof that the boundaries of all escape domains of level k ≥ 3 (these escape domains are

called Sierpinski holes) are Jordan curves relies on three ingredients: the boundary regularityof ∂H0, holomorphic motion and continuity of cut rays. We remark that our approach alsoapplies to ∂H2. This will yield a different proof from Devaney’s in [4].

§6 Acknowledge

This survey is based on the authors’ joint work with Professors Weiyuan Qiu and PascaleRoesch. The authors would like to thank them for discussions and comments.

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Department of Mathematics, Zhejiang University, Hangzhou 310027, China.Email: [email protected] (X.Wang), [email protected] (Y. Yin).

dynamics, topology and bifurcations of mcmullen maps

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