Tessellation and Lyubich-Minsky laminations
associated with quadratic maps II:
Topological structures of 3-laminations
Tomoki Kawahira
Graduate School of Mathematics, Nagoya University

September 15, 2008
Abstract
According to an analogy to quasi-Fuchsian groups, we investigate the topological andcombinatorial structures of Lyubich and Minsky’s aﬃne and hyperbolic 3-laminations
associated with hyperbolic and parabolic quadratic maps.
We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. This gives a good reason to regard
the main cardioid of the Mandelbrot set as an analogue of the Bers slices in the quasiFuchsian space. Then we describe the topological and combinatorial changes of laminations associated with hyperbolic-to-parabolic degenerations (and parabolic-to-hyperbolic
bifurcations) of quadratic maps. For example, the diﬀerences between the structures of
the quotient 3-laminations of Douady’s rabbit, the Cauliﬂower, and z → z 2 are described.
The descriptions employ a new method of tessellation inside the ﬁlled Julia set introduced in Part I [Ka3] that works like external rays outside the Julia set.
The 2000 Mathematics Subject Classiﬁcation: 37F45, 37F99.

Contents
1 Introduction

2

2 Lyubich-Minsky laminations

4

3 Stability of hyperbolics

10

4 Analogy to quasi-Fuchsian deformations

14

5 Degeneration pairs and Tessellation18

6 Degeneration of the aﬃne laminations

27

7 Degeneration of the H3 -laminations

37

8 Degeneration of the quotient 3-laminations

43

9 Degeneration of the lower ends

48

10 Twisting the lower ends

56

A Appendix: Remarks on Hénon mappings

66

1

1

Introduction

As an analogue of the hyperbolic 3-orbifolds associated with Kleinian groups, Lyubich andMinsky [LM] introduced hyperbolic orbifold 3-laminations associated with rational maps.
For a rational map f : C̄ → C̄, there exist three kinds of laminations Af , Hf , and Mf as
following:• The aﬃne lamination Af , a Riemann surface lamination with leaves isomorphic to C
or its quotient orbifold.
• The H3 -lamination Hf , a hyperbolic 3-lamination with leaves isomorphic to H3 or its
quotient orbifold. (This is a 3D-extension of the aﬃne lamination of f .)As a Kleinian group acts on C̄ and H3 , there is a natural homeomorphic and leafwiseisomorphic action fˆ on Af and Hf induced from that of f . In particular, the action isproperly discontinuous on Hf . The third lamination is:• The quotient 3-lamination Mf := Hf /fˆ , whose leaves are hyperbolic 3-orbifolds.
In addition, we have the Fatou-Julia decomposition Ff  Jf = Af of the aﬃne lamination.
As a Kleinian group acts on the complement of the limit set properly discontinuously, fˆ
acts on Ff properly discontinuously. Hence the quotient Ff /fˆ forms a Riemann surface
lamination and we regard it as the conformal boundary of Mf . In this paper we say thequotient Mf = (Hf ∪ Ff )/fˆ the Kleinian lamination.
Lyubich and Minsky applied analogous arguments on rigidity theorems of hyperbolic 3orbifolds to the hyperbolic 3-laminations, and showed a rigidity result of rational maps thathave no recurrent critical points or parabolic points [LM, Theorem 9.1]. However, evenfor hyperbolic quadratic maps like z 2 − 1 or Douady’s rabbit, the precise structure of theirlaminations have not been investigated. (See the questions in [LM, §10].) Moreover, even forthe simplest parabolic quadratic map z 2 + 1/4 (“the Cauliﬂower”) whose 3-lamination has a
cuspidal part, its structure had not been precisely known. The aim of this paper is to give amethod to describe the topological and combinatorial changes of laminations associated withthe motion of parameter c of z 2 + c from one hyperbolic component to another via parabolic
parameters. In particular, our investigation of the Kleinian laminations is explained in termsof the analogy between the Bers slices of the quasi-Fuchsian deformation and the Mandelbrot
set.
Outline of the paper. Section 2 is devoted for the constructions of the Lyubich-Minsky
laminations according to [LM]. Then we prove that the aﬃne lamination Af of an expansive (parabolic) rational map f (that is, the Julia set contains no critical point) is actually
identiﬁed as a proper subset of the natural extension (projective limit) Nf of f .
In Section 3, we consider perturbation of the rational maps with superattracting cycles.
We will show that we can relax such cycles to attracting cycles without changing the geometryof Anf . (In our terminology, attracting cycles are not superattracting.) As a corollary, alllaminations associated with hyperbolic rational maps in the same hyperbolic component are
quasiconformally (or quasi-isometrically) equivalent.
The analogy between the Bers slices of the quasi-Fuchsian deformation space and theMandelbrot set is explained in Section 4 in the lamination context. Our stance is that thequadratic map fc (z) = z 2 + c is a deformation of f0 (z) = z 2 , whose Kleinian lamination isa product of Sullivan’s solenoidal Riemann surface lamination S0 and a closed interval [0, 1].
By using the results in the Section 3, we will show that all fc with c in the main cardioid of
the Mandelbrot set have quasi-isometrically the same Kleinian laminations. We also deﬁne2

the lower and upper ends of the quotient 3-laminations which correspond to the two ends ofthe Kleinian manifolds of quasi-Fuchsian groups.
Then we go back to the settings of Part I [Ka3] and study degeneration and bifurcation
processes of quadratic laminations.
In Section 5, we summerize the results and notation given in Part I [Ka3]. We formalize
hyperbolic-to-parabolic degeneration of quadratic maps in terms of degeneration pairs (f →g) and we recall some properties of tessellation of the interior of the ﬁlled Julia sets associatedwith (f → g).
From this section, we often use the following standard examples to explain the ideas:• The Cauliﬂowers: The family {z 2 + c} with 0 < c ≤ 1/4, which gives the simplest
degeneration process.
• The Rabbits: The family {z 2 + c} with parameter c moving from 0 to the center ofthe 1/3-limb (Douady’s rabbit, with the value denoted by crab ) of the Mandelbrot set.
(See Figure 1 of Part I [Ka3].) This is a typical degeneration and bifurcation process.
• The Airplanes: The family {z 2 + c} with cair < c ≤ −1.75 and cair = −1.7548 · · · ,which gives a real superattracting cycle of period three. This is a typical degeneration
process to the root of a primitive copy of the Mandelbrot set.
In Section 6, we consider hyperbolic-to-parabolic degenerations of quadratic natural extensions and aﬃne laminations. For hyperbolic f and parabolic g with f → g, we ﬁrst lift
the pinching semiconjugacy h on the sphere constructed in the ﬁrst part of this work ([Ka3])
to their natural extensions; ĥ : Nf → Ng . Then we describe the structure of Ag as a degeneration of Af by using ĥ. The conclusion here is that as f → g the topology of Af is leafwisepreserved to Ag except on ﬁnitely many leaves, called the principal leaves.
In Section 7, we extend the semiconjugacy ĥ on the aﬃne lamination to the H3 -lamination.
Though it is impossible to extend it to the whole Hf , we can ﬁnd the best possible extension
which is enough to describe the structure of Hg . We will show that the leafwise topology isalso preserved as f → g except on the 3D extension of the principal leaves.
In Section 8, we construct a pinching map on the quotient 3-laminations by using thesemiconjugacy above. This pinching map will give a topological picture of Mg based on thatof Mf . We will see that the topology of each leaf of Mf is preserved as f → g, but thegeometry changes at the quotient principal leaf.
The lower ends in the conformal boundaries better reﬂect this geometric change. In
Section 9, we investigate the structures of the lower ends of the Kleinian laminations Mf
and Mg by checking how tessellations are lifted to the conformal boundaries. Then we willconclude that Mf cannot be homeomorphic to Mg . As an application, we describe theKleinian laminations of the Cauliﬂowers (§9.5) in detail.
In Section 10, we consider the topological changes of the lower ends associated withbifurcations. More precisely, we take two distinct degeneration pairs (f1 → g1 ) and (f2 → g2 )with g1 = g2 and we describe the combinatorial diﬀerence between the lower ends of Mf1
and Mf2 by means of a Dehn-twist-like operation. For instance, we will conclude:
Theorem 1.1 The Kleinian laminations of f0 : z → z 2 , the Cauliﬂower f1/4 : z → z 2 + 1/4,
and Douady’s Rabbit are not homeomorphic each other.
See §10.5 for more detailed description of the Kleinian lamination of Douady’s rabbit. We
will also give a result which supports a geometric analogy between the Bers slices in thequasi-Fuchsian deformation spaces and the main cardioid of the Mandelbrot set:3

Theorem 1.2 For any fc (z) = z 2 + c with c in the Mandelbrot set, its Kleinian lamination
Mfc has a boundary component that is conformally the same as S0 . It also has a productstructure ≈ S0 × [0, 1] if and only if c is in the main cardioid of the Mandelbrot set.
See §4.2 and Theorem 10.6.
Hénon mappings with small Jacobian.
There is another viewpoint of the resultsin Section 6. In [HO], Hubbard and Oberste-Vorth showed a strong connection between
the projective limits of hyperbolic polynomials and the Julia sets of Hénon mappings withsmall Jacobian. Since Lyubich-Minsky laminations are based on the projective limits ofrational maps (the natural extension in their/our terminology), our method automaticallygives topological description of the Julia sets of some quadratic Hénon mappings with small
Jacobian. The last section (Appendix A) of this paper is a brief survey on relations between
the Julia sets of Hénon mappings and the aﬃne laminations.
Note. Most of pictures in this paper are colored.
Acknowledgments. I would like to thank C. Cabrera, K. Ito, M. Lyubich, and Y. Tanaka
for stimulating discussions, and the referee for comments. M. Lyubich also gave me opportunities to visit SUNY at Stony Brook, University of Toronto, and the Fields Institute wherepart of this work was being prepared. I also would like to thank the Fields Institute andthe Institute des Hautes Études Scientiﬁques for their hospitality while this work was being
revised.
This research is partially supported by JSPS Research Fellowships for Young Scientists,
JSPS Grant-in-Aid for Young Scientists, Nagoya University, the Circle for the Promotionof Science and Engineering, Inamori Foundation, and the IHÉS. I sincerely appreciate theirsupports.

2

Lyubich-Minsky laminations

In this section we sketch Lyubich and Minsky’s universal construction of the aﬃne, H3 -,and quotient laminations associated with general rational maps in [LM, §6 and §7]. (See[L] for a brief survey by Lyubich. [KL, §3] would be helpful also. For the general theory oflaminations, see [CC]. See also [MS, §6] and [S].)Furthermore, for any expansive (= parabolic) rational map f , we will show that its aﬃnelamination Af is identiﬁed as its aﬃne part Anf , which is an analytically well-behaved partof the natural extension. In particular, for hyperbolic and parabolic quadratic maps f andg, we may regard Af and Ag as Anf and Ang .

2.1

The universal orbifold laminations

Let U be the set of all non-constant meromorphic functions on the complex plane C, with
topology induced by the uniform convergence on compact sets. Let us consider the productspace Û := U × U × · · · . We consider each element ψ̂ = (ψ0 , ψ−1 , . . .) ∈ Û as a map from C
to C̄ × C̄ × · · · :ψ̂ : C → C̄ × C̄ × · · ·w → (ψ0 (w), ψ−1 (w), . . .).
By taking the value at w = 0 of this map, we have the following continuous map:
℘ : Û → C̄ × C̄ × · · ·ψ̂ → (ψ0 (0), ψ−1 (0), . . .).
4

Right actions and quotient universal foliations. Let Aﬀ be the set of complex aﬃnemaps on C. Then we may identify Aﬀ as A := C × C∗ by regarding δ(w) = a + bw as(a, b) ∈ A. We also regard the hyperbolic 3-space H3 as C × R+ . Note that we have naturalprojections from A over H3 and from H3 over C, given by
A

−→

H3

−→ C

(a, b) −→ (a, |b|) −→ a.
For δ ∈ Aﬀ and ψ̂ ∈ Û , the composition ψ̂ ◦ δ is also an element of Û . Thus δ ∈ Aﬀ actson the right and the orbits {ψ̂ ◦ Aﬀ : ψ̂ ∈ Û } form a foliation of Û with leaves isomorphic to
A or its quotient orbifold. ([LM, Lemma 7.1]). More precisely, a natural map
A(a, b) −→ ψ̂(a + bw) ∈ ψ̂ ◦ Aﬀ ⊂ Ûgives a branched chart for the orbit ψ̂ ◦ Aﬀ in general. For example, consider the orbit ψ̂ ◦ Aﬀ
2
2
of ψ̂(w) = (ew , ew , . . .). Then ψ̂(a + bw) and ψ̂(−a − bw) determine the same point in Û
and the chart have singularities over a = 0.
Let us deﬁne the quotient universal foliations Û h and Û a so that there exist projections
Û −→ Û h −→ Û aanalogous to the projections A → H3 → C above. We consider the following two equivalence
relations in Û :1. ψ̂ ∼h ψ̂  if there exists a t ∈ R such that ψ̂(w) = ψ̂  (e2πit w) for all w ∈ C.
2. ψ̂ ∼a ψ̂  if there exists a b ∈ C∗ such that ψ̂(w) = ψ̂  (bw) for all w ∈ C.
Let [ψ̂(w)]h and [ψ̂(w)]a denote the equivalent classes of ψ̂ for ∼h and ∼a , with a dummy
variable w. Then we have a natural projection pr : [ψ̂(w)]h → [ψ̂(w)]a .
For an orbifold leaf ψ̂ ◦ Aﬀ ∈ Û, the quotients (ψ̂ ◦ Aﬀ)/∼h and (ψ̂ ◦ Aﬀ)/∼a may havemore singularities in general and they form 3- and 2-orbifolds. In fact, since the following
diagrams commute, these quotients can be considered as hyperbolic 3- and aﬃne orbifolds
respectively:(a, b) −−−−→ ψ̂(a + bw)
A −−−−→ψ̂ ◦ Aﬀ





∼

∼

 h

 h
H3 −−−−→ (ψ̂ ◦ Aﬀ)/∼h



pr



(a, |b|) −−−−→ [ψ̂(a + |b|w)]h



pr



C −−−−→ (ψ̂ ◦ Aﬀ)/∼aa−−−−→ [ψ̂(a + w)]a .

We say the branched charts H3 → (ψ̂ ◦ Aﬀ)/∼h and C → (ψ̂ ◦ Aﬀ)/∼a are the orbifold chartswith respect to ψ̂.
Now we set Û h := Û /∼h and Û a := Û/∼a . It is known that they are 3- and 2-orbifold
foliations ([LM, Corollary 7.3]). Note that the continuous map ℘ still makes sense from Û hor Û a to C̄ × C̄ × · · · .
In this paper, we will mainly work with non-singular cases: That is, the quotients of theorbit ψ̂◦Aﬀ are isomorphic to H3 or C. For (a, b) ∈ A, we say [ψ̂(a+bw)]h (resp. [ψ̂(a+bw)]a )has coordinate (a, |b|) ∈ H3 (resp. a ∈ C) with respect to ψ̂.

5Left actions and the global attractor. For a rational map f and ψ ∈ U, f ◦ ψ is also anelement of U. Thus f acts on U from the left and we denote this action fu : ψ → f ◦ ψ. Notew+πithat this action is not generally injective. For example, if f (z) = z 2 then fu (ew ) = fu (e n ).
Let us consider a closed invariant set Kf of the semigroup fu deﬁned by n≥0 fu (U ),which we call the global attractor [LM, §§7.3]. To obtain a homeomorphic action over Kf , weconsider the natural extension
K̂f := ψ̂ = (ψ0 , ψ−1 , . . .) ∈ Û : ψ0 ∈ Kf , fu (ψ−n ) = ψ−n+1
with topology induced from Û. On this set fu is lifted to a homeomorphic actionfˆu : K̂f(ψ0 , ψ−1 , . . .) → (fu (ψ0 ), ψ0 , ψ−1 , . . .) ∈ K̂fwhich is a leafwise isomorphism. It is also known that K̂fh := K̂f /∼h and K̂fa := K̂f /∼a are
3- or 2-orbifold laminations [LM, Lemma 7.4]. (In [KL], the set K̂fa is denoted by Af andcalled the universal aﬃne lamination.) Moreover, the action of fˆu descends to well-deﬁned
actions fˆa and fˆh on K̂fh and K̂fa as following:fˆh ([ψ̂]h ) := [fˆu (ψ̂)]hand fˆa ([ψ̂]a ) := [fˆu (ψ̂)]a .

One can easily check that they send a leaf to a leaf isomorphically.

2.2

The aﬃne and hyperbolic laminations

Natural extension. For the dynamics of rational f , the set of all possible backward orbits
Nf := ẑ = (z0 , z−1 , . . .) : z0 ∈ C̄, f (z−n−1 ) = z−n
is called the natural extension of f , which is equipped with a topology from C̄ × C̄ × · · · [LM,§3]. For ẑ = (z0 , z−1 , . . .) ∈ Nf , the lift fˆn of f and the natural projection πf are deﬁned byfˆn (ẑ) := (f (z0 ), f (z−1 ), . . .) = (f (z0 ), z0 , z−1 , . . .) andπf (ẑ) := z0 .
It is clear that fˆn is a homeomorphism, and satisﬁes πf ◦ fˆn = f ◦ πf .
Remarks on notation.
• For simplicity, we will abuse fˆ to denote fˆu , fˆh , fˆa , and fˆn when it does not lead
confusion.
• We will use x̃ or ỹ to denote points in K̂fh , and x̂ or ŷ to denote points in K̂fa . The
points in the natural extension Nf are denoted by ẑ or ζ̂, etc.
Regular leaf space and its aﬃne part. An element ẑ = (z0 , z−1 , . . .) ∈ Nf is regular if
there exists a neighborhood U0 of z0 such that its pull-backs
z0 ∈ U0 ← U−1 ← U−2 ← · · ·along the backward orbit ẑ are eventually univalent. For example, backward orbits remaining
in an attracting or parabolic cycle are not regular, since at least one critical orbit accumulates
6

on such a cycle. We say that such backward orbits form an attracting or parabolic cycle in
Nf .
Let Rf denote the set of all regular points in Nf . We call Rf the regular leaf space of f[LM, §§3.2]. A leaf of Rf is a path-connected component of Rf . Note that Rf is invariant
under fˆ. It is known that each leaf of Rf has a natural complex structure of non-compact
Riemann surface induced from the dynamics downstairs. We take the union of all the leaves
isomorphic to C, and call it the aﬃne part Anf of Rf [LM, §§4.2]. By Lemma 3.1, Proposition
4.5, Lemma 4.8 of [LM], we have
Proposition 2.1 (Aﬃne part) The aﬃne part Anf satisﬁes the following properties:(1) The map fˆ acts on Anf homeomorphically. Moreover, for any leaf L in Anf , fˆ : L → fˆ(L)is an isomorphism.
(2) Any leaf L of Anf is dense in Nf .
(3) If all critical points of f are non-recurrent,
Anf = Rf = Nf − {attracting and parabolic cycles}.
Note that we may regard the action fˆ : L → fˆ(L) an aﬃne map by (1), since both L andfˆ(L) are isomorphic to C.
We will apply this proposition to hyperbolic and parabolic quadratic maps in Section 6.
Aﬃne lamination. Take a point ẑ = (z0 , z−1 , . . .) ∈ Anf and the leaf L = L(ẑ) whichcontains ẑ. Then there is a uniformization φ : C → L with φ(0) = ẑ. (Such a φ is not unique.)One can easily check that the sequence of maps {ψ−N = πf ◦ fˆ−N ◦φ : C → C̄}N ≥0 determines
an element ψ̂ = (ψ−N )N ≥0 in K̂f with ℘(ψ̂) = ẑ. Moreover, the element x̂ = [ψ̂]a in K̂fa does
not depend on the choice of φ above and is uniquely determined. Let ι = ιf : Anf → K̂fa
denote the inclusion deﬁned by ι(ẑ) = x̂. Note that ι is injective and ℘ ◦ ι = id on Anf . For
later use we set π̂f := πf ◦ ℘ : K̂fa → C̄.
Now we claim that ι sends L = L(ẑ) above to a leaf L̂ = L̂(x̂) of K̂fa isomorphic to C: In
fact, for any ζ̂ ∈ L, there exists a unique a ∈ C such that ζ̂ = φ(a) thus ζ̂ and the element of
the form [ψ̂(a + bw)]a (b ∈ C∗ ) in K̂fa have one-to-one correspondence. This means that thequotient orbit (ψ̂ ◦ Aﬀ)/∼a is precisely parameterized by C and ι(ζ̂) has coordinate a withrespect to ψ̂. We say φ̂ := ι ◦ φ is a uniformization of L̂ with respect to ψ̂.
Let Af denote ι(Anf ) in K̂fa . Then Af consists of leaves isomorphic to C and is invariant
under fˆ. Now we take the closure Af of Af in K̂fa and call it the aﬃne lamination of f .
There may be some additional orbifold leaves in Af − Af in general. However, in the casewhere f is hyperbolic or parabolic (quadratic) maps, we will show that Af is already closed
in K̂fa , and thus Af = Af (Proposition 2.2).
H3 -lamination. For x̂ ∈ Af and one of its representative ψ̂ ∈ K̂f (that is, x̂ = [ψ̂]a ), thequotient orbit (ψ̂ ◦ Aﬀ)/∼h is a leaf of K̂fh with orbifold chart (a, |b|) → [ψ̂(a + |b|w)]h withrespect to ψ̂. Now we gather such leaves and deﬁne the H3 -lamination associated with f as
following:
Hf := {ψ̂ ∈ K̂f : [ψ̂]a ∈ Af } / ∼hThis is one of our hyperbolic 3-orbifold laminations. Note that the H3 -lamination is givenby taking a suitable R+ -bundle of the aﬃne leaves with vertical projection pr : Hf → Af .
7

Indeed, Hf can be geometrically constructed as an “scaling bundle” of Af [KL]. But thevirtue of our universal construction is that Hf is a sublamination of the universal metrized
space Û h for any rational map f .
In this paper, for a subset X of Af , we denote its ﬁber pr−1 (X) by X h . (In [KL], such anh
X is denoted by HX and they say H : X → X h is the hyperbolization functor.) For example,
for a leaf L̂ of Af , L̂h is a leaf of Hf with pr(L̂h ) = L̂. For the orbifold chart φ̂ : C → L̂with respect to a ﬁxed ψ̂ ∈ K̂f , let φ̂h denote the orbifold chart φ̂h : H3 → L̂h with respectto ψ̂. Then pr : L̂h → L̂ is consistent with the vertical projection from H3 to C; that is,pr ◦ φ̂h (a, |b|) = φ̂(a).
Quotient 3-lamination. Now Hf is invariant under the action fˆ = fˆh over K̂fh . It is known
that the action of the cyclic group fˆ on Hf is properly discontinuous ([LM, Proposition6.2]). Thus the quotient Mf := Hf /fˆ is Hausdorﬀ, and inherits the laminar structure of
Hf . We call Mf the quotient 3-lamination of f . This is our second hyperbolic 3-lamination,
which corresponds to the hyperbolic 3-orbifold associated with a Kleinian group.
Since fˆ carries leaves of Hf isometrically, a leaf  of Mf is isomorphic to an hyperbolic
3-orbifold. If a leaf L̂h of Hf is periodic, i.e. fˆn (L̂h ) = L̂h for some n ∈ Z, then  = L̂h /fˆ
is either a solid torus or a rank one cusp; or it has a ﬁnite branched cover from H3 . If L̂h is
not periodic, the quotient is isomorphic to H3 . In [KL, Prop.1.22] one can ﬁnd a complete
classiﬁcation of such leaves.
Figure 1 is a diagram showing the relation between the objects we have deﬁned so far.

Figure 1: The objects in the dashed square are deﬁned by means of the universal laminations.
On the right Γ is a Kleinian group acting on H3 .
Conformal boundary and Kleinian lamination. According to the Fatou-Julia decomposition C̄ = Ff  Jf of the dynamics on C̄, we have a natural decomposition
Af = Ff  Jf := π̂f−1 (Ff )  π̂f−1 (Jf )of the aﬃne lamination. Both Ff and Jf are invariant under fˆ. In particular, fˆ acts on
Ff properly discontinuously [LM, §§3.5]. The quotient ∂Mf := Ff /fˆ is a Riemann surface
lamination (possibly with many components) and called the conformal boundary of Mf . The
union Mf = Mf ∪ ∂Mf forms a 3-lamination with boundary. We call Mf the quotient 3lamination with conformal boundary, or simply, the Kleinian lamination of f after “Kleinan
manifolds” in the theory of hyperbolic 3-manifolds.

8

2.3

Aﬃne laminations for expansive rational maps.

We say a rational map f is expansive (or parabolic) if any two distinct orbits in the Julia
set are eventually a deﬁnite distance away from each other. By Denker and Urbaǹski [DU],
it is shown that f is expansive if and only if the Julia set Jf contains no critical points.
Equivalently, every critical point of f is attracted to an attracting or parabolic cycle. Notethat every hyperbolic (= expanding) map is a special case of expansive map according to thisterminology. Here we will establish:
Proposition 2.2 For any expansive rational map f , the set Af is closed in K̂fa , and thus
Af = Af .
Before the proof, let us consider the case of quadratic map f = fc : z → z 2 + c. A direct
corollary of this proposition is:
Corollary 2.3 If f = fc has an attracting or parabolic cycle in C, then the aﬃne lamination
Af is equal to Af .
Thus we can identify Anf in the natural extensions (or in C̄ × C̄ × · · · ) as Af in the universal
aﬃne laminations (or in Û a ). More informally, we may consider that ẑ is just another nameof x̂ = ιf (ẑ), and vice versa.
Proof of Proposition 2.2.
Take any sequence x̂n ∈ Af with x̂n → x̂ ∈ Af . Setẑn = (zn,−N )N ≥0 := ℘(x̂n ) and ẑ = (z−N )N ≥0 := ℘(x̂). By continuity of ℘, we have ẑn → ẑin Anf . Set ŷ := ι(ẑ) ∈ Af . Now it is enough to show that x̂n → ŷ in Af and thus ŷ = x̂. If
f is hyperbolic, this is straightforward by applying [LM, Propositions 7.5 and 8.2]. If f has
parabolic cycles, we need some modiﬁcation.
Backward invariant set Ω. Let O be an attracting or parabolic cycle. If O is attracting,
we take an open set EO as a neighborhood of O with f (EO ) ⊂ EO . If parabolic, we take
EO as attracting petals attached to each point of O such that f (EO ) − O ⊂ EO . Let E
be the union of such EO for all attracting and parabolic cycles. Since f is expansive, there
exists an integer M > 0 such that the compact set Ω := C̄ − f −M (E) contains no critical or
postcritical point except parabolic cycles. Note that f −1 (Ω) − (parabolic cycles) ⊂ Ω◦ andthus every backward orbit except attracting or parabolic cycles is eventually trapped into Ω◦
and accumulates on the Julia set.
Convergence on any large disks. For ρ > 0, let Dρ denote the disk centered at 0 withradius ρ. Let us ﬁx a representative ψ̂ = (ψ−N )N ≥0 ∈ K̂f of ŷ. We claim that for arbitrarily
large N0 > 0 and ρ > 0, we can choose a representative ψ̂n = (ψn,−N )N ≥0 ∈ K̂f of x̂n suchthat ψn,−N0 +m converges to ψ−N0 +m uniformly on Dρ for m = 1, . . . , N0 .
Take a suﬃciently large ρ > 0 such that = ρ/ρ  1, and set V := Dρ . We ﬁrst claimthat for all N  0, V−N := ψ−N (V)  Ω. Set L := L(ẑ) in Anf , the leaf containing ẑ. Then
there exists a unique uniformization φ : L → C such that ψ0 = f N ◦ψ−N = f N ◦(πf ◦ fˆ−N ◦φ).
Since deg(ψ0 |V ) is constant and ﬁnite, so is deg(f N ◦ πf ◦ fˆ−N ◦ φ) = deg(f N ◦ πf ) for all
N  0. Hence f : V−N → V−N +1 is univalent for all N  0. This implies that the backward
sequence of open sets V0 ← V−1 ← · · · does not contain the backward orbits corresponding
to the invariant lift of the attracting or parabolic cycles. Since V0 is compact, eventually V−N
is contained in Ω◦ for all N  0. Note that for such an N , the map πf ◦ fˆ−N : φ(V) → V−N
gives a local chart of L around ẑ (cf. [LM, Lemma 3.1]) and it implies that ψ−N : V → V−N
is univalent for all N  0.
9

For each n, take a representative ψ̂n = (ψn,−N )N ≥0 ∈ K̂f with ℘(ψ̂n ) = ẑn . Fix a large
N ≥ N0 so that V−N  Ω. Since ẑn converges to ẑ in Anf , we have zn,−N ∈ V−N for alln  0. Thus any pull-back of V−N along ẑn is univalent for all n  0. Now we can apply theargument of [LM, Proposition 7.5]: Since each ψn,−N is also induced from a uniformizationof a leaf L(ẑn ) of Anf , there exists a neighborhood Vn ⊂ C of 0 such that ψn,−N maps Vn
univalently onto V−N . Consider a univalent map χn = χn,−N : V → Vn deﬁned by the−1◦ ψ−N via V−N . Then we have ψ−N +m = ψn,−N +m ◦ χn on V for
univalent branch of ψn,−N
all m = 0, 1, . . . , N .
Let an be the point in V with χn (an ) = 0. Set χ̃n (w) := χn (w + an ) which is deﬁned
on the slided disk V − an . By normalizing ψ̂n such that (ψ−N ) (an ) = (ψ̂n,−N ) (0), we have−1−1χ̃n (0) = 0 and (χ̃n ) (w) = 1. Since an = ψ−N(zn,−N ) tends to ψ−N(z−N ) = 0, we haveDρ − an  D2ρ  Dρ /2  V − anfor all n  0. By applying the Koebe distortion theorem to χ̃n on D2ρ  Dρ /2 , we haveχ̃n (w) = w(1 + O( )) on D2ρ and thus χn (w) = (w − an )(1 + O( )) on Dρ . This implies thatχn → id uniformly on Dρ as n → ∞, and thus ψn,−N +m → ψ−N +m for all m = 0, 1, . . . , Non Dρ . We conclude that x̂n converges to x̂ = ŷ within Af .
Tameness of the laminations. After [KL, §3.1.7], we say a lamination is tame if it islocally compact. Since Û a and Û h are not locally compact, the tameness of Af or Hf ofexpansive f is not obvious. However, since Anf and Af have the same topology and Anf islocally compact, we have:
Corollary 2.4 (Tameness) The aﬃne lamination Af of the expansive rational map f istame. Hence the hyperbolic 3-laminations Hf and Mf are tame too.
The local compactness of the quotient 3-lamination may play an important role when one
try to extend Lyubich and Minsky’s rigidity result of the critically non-recurrent maps withno parabolic points [LM, Theorem 9.1] in the lamination context. However, Haı̈ssinsky [Ha]extended their theorem without using the lamination theory to wider class of rational maps
called (uniformly) weakly hyperbolic, which may have parabolic cycles. For more results on
the rigidity, see references in [Ha].

3

Stability of hyperbolics

In this section we check the topological/geometric stability of laminations associated withhyperbolic rational maps with superattracting cycles. In our quadratic case, it is justiﬁedthat the laminations of f = fc in a hyperbolic component and those of its hyperbolic center
have topologically the same structures.
The key result is that we can perturb superattracting cycles into attracting cycles withoutchanging most part of the dynamics (Theorem 3.2).
Leaf spaces. It is convenient to generalize the notion of lamination slightly. For a given
integer k ≥ 1, we say a topological space L is a leaf space if it is a connected topologicalspace that consists of path-connected components (“leaves”) homeomorphic to k-orbifolds.
For example, the regular leaf space Rf , the aﬃne part Anf , and the aﬃne lamination Af of
a rational map f are leaf spaces with leaves isomorphic to Riemann orbifolds (= complex
1-orbifolds). (Technically, we have to remove “isolated leaves” from Af when f is Chebyshev
or Lattès. See [LM, §5.4] and [KL, §3.1.1]. We do not deal with these cases in this paper.)10

Quasi-isometry and quasiconformal maps. Let L1 and L2 be leaf spaces with leaves
isomorphic to hyperbolic 3-orbifolds (resp. Riemann orbifolds). Suppose that there is ahomeomorphism H : L1 → L2 . We say H is (leafwise) quasi-isometric (resp. quasiconformal)
if there is a uniform K ≥ 1 such that H sends each leaf in L1 to a leaf in L2 as a K-quasiisometry (resp. K-quasiconformal map).
In addition, if L1 and L2 support dynamics F1 : L1 → L1 and F2 : L2 → L2 which
are conjugated by H, we say L1 and L2 are quasi-isometrically (resp. quasiconformally)equivalent.
The following proposition is given in the proof of [LM, Theorem 9.1, Lemma 9.3]:
Proposition 3.1 (Quasi-isometric extension) For rational maps f and g, suppose that
there exists a quasiconformal conjugacy ĥa : Af → Ag . Then it is lifted to their H3 laminations ĥh : Hf → Hg as a quasi-isometric conjugacy. Moreover, it descends to aquasi-isometric homeomorphism ĥq : Mf → Mg , which extends to ĥq : Mf → Mg .
Perturbation of superattractings. Let Ratd denote the space of rational maps of degreed ≥ 2. For a given rational map f ∈ Ratd , we consider a perturbation f → f ( → 0) within
Ratd with respect to the topology deﬁned by uniform convergence (equivalently, convergenceof coeﬃcients).
Here is the main theorem of this section:
Theorem 3.2 (Relaxing superattracting cycles) Suppose f has a superattracting cycle
O. Then there exists a perturbation f → f with the following properties:• f → f just relaxes O to be an attracting cycle and preserve the topological dynamics
except the immediate basin of O; and• The aﬃne parts Anf and Anf are quasiconformally equivalent.
Assuming this theorem, we can prove:
Theorem 3.3 (Stability of hyperbolics) Let f be a hyperbolic rational map. For any
small perturbation f → f , their aﬃne, H3 -laminations are quasiconformally or quasi-isometricallyequivalent. In particular, their Kleinian laminations are quasi-isometrically homeomorphic.
Proof. If f has no superattracting cycle and  1, it is known that we have a quasiconformal conjugacy between any perturbation f and f . It naturally extends to a quasiconformal
conjugacy on the aﬃne parts Anf and Anf .
If f has superattracting cycles, by Theorem 3.2, we can perturb f within any smallneighborhood to relax all of the superattractings. We can do the same for f in the sameneighborhood, and thus there exists a quasiconformal conjugacy on the aﬃne parts Anf andAnf .
Since Anf and Anf are quasiconformally equivalent, the aﬃne laminations Af and Af arealso quasiconformally equivalent by Proposition 2.2. (For example, ιf gives the conformal
conjugacy between Anf and Af .) By Proposition 3.1, this perturbation is also accompanied
by quasi-isometries between their H3 - and quotient 3-laminations. Since the quasiconformal conjugacy between the aﬃne laminations preserves the Fatou-Julia decomposition, theisometry between the quotient laminations extends to the conformal boundaries.


11

Thus hyperbolic rational maps are stable in the sense of Lyubich-Minsky laminations.
Here is an interesting question for people familiar with J-stability of holomorphic family ofrational maps [MSS]: Is J-stability of a rational map equivalent to quasiconformal stabilityof its aﬃne lamination?
In our quadratic case, we have:
Corollary 3.4 Let X ⊂ C be a hyperbolic component of the Mandelbrot set. For any c and c
in X, their aﬃne and hyperbolic 3-laminations are quasiconformally and quasi-isometricallyequivalent respectively. Moreover, their quotient laminations are quasi-isometrically homeomorphic including their conformal boundaries.
Independently C.Cabrera [Ca] proved the topological equivalence of aﬃne laminations forc and c as above. A remarkable fact is that he also proved the converse. By the corollary
above, we can restate it as a rigidity theorem:
Theorem 3.5 (Rigidity by Cabrera) If there exists an orientation preserving homeomorphism between the aﬃne laminations of hyperbolic fc and fc , then c and c are in the same
hyperbolic component. Moreover, they are actually quasiconformally equivalent.
See [Ca] for the proof. It seems likely that we can replace “aﬃne laminations” by “H3 laminations”. However, by the investigation in this paper, it seems unlikely that we canreplace it by the “quotient 3-laminations”. (It is more likely that we can replace it by“Kleinian laminations”.)
Proof of Theorem 3.2. We construct such a perturbation by quasiconformal perturbationof f . By taking a Möbius conjugation, we may assume that 0 is a critical point belonging to
O. Let l be the period of O and α be a point in O such that f (α) = 0. Note that α may bea critical point too, like ∞ of f (z) = 1/z 2 .
Let us ﬁx a small λ > 0 and set Dλ := {w ∈ C : |w| < λ}. Then there exists a diskneighborhood D of 0 and the Böttcher coordinate Φ : D → Dλ such that Φ : z → w
conformally conjugates f l on D to w → wp for some p ≥ 2. Let D be the component off −1 (D) containing α. If necessary, we use smaller λ so that D − {α} contains no criticalpoint. Note that Φ ◦ f : D → Dλ semi-conjugates f l on D and w → wp on Dλ .
Local perturbation. Next we perturb f : D → D as following: Fix ρ1 and ρ2 such that
λp < ρ1 < ρ2 < λ, and take any > 0 satisfying ρ1 + < ρ2 . Now we deﬁne a homeomorphismH : Dλ → Dλ byw+
if |w| < ρ1ρ2 − |w|
H (w) :=w+ ·if ρ1 ≤ |w| ≤ ρ2
ρ2 − ρ1w
if ρ2 < |w| < λ.
One can check that this map is actually a homeomorphism. See Figure 2 for the idea. Notethat the L∞ -norm of the Beltrami diﬀerential of H is O( ).
Quasiregular map. Now we deﬁne a branched covering g : C̄ → C̄ by:g (z) :=Φ−1 ◦ H ◦ Φ ◦ f (z) if z ∈ Df (z)if z ∈/ D .

Then the following holds:• g → f as

→ 0, since H → id.

12

Figure 2: This diagram illustrates the idea of the local perturbation. Note that f and Φ ◦ f inthis diagram may have the degree two or more. To get the sliding map H , consider a cone ofheight 1 based on Dρ2 . Then slide the summit without changing the base. By truncating thiscone at the level of (ρ2 − ρ1 )/ρ2 , we obtain a sliding map of a truncated cone. Via verticalprojections and this sliding map, we obtain H above.
• g is quasiregular. Indeed, it is holomorphic except on a compact annulus A :={z ∈ D  : ρ1 ≤ Φ ◦ f (z) ≤ ρ2 }.
• gl : D → D is viewed as w → wp + through Φ. Hence 0 ∈ D is not periodic of periodl.
The almost complex structure. Let σ0 be the standard complex structure of C̄. We
deﬁne a g -invariant almost complex structure σ byσ :=(gn )∗ σ0 on g−n (D), n ≥ 0otherwise.
σ0

Since any orbit passes A (where gn gains dilatation) at most once, we can integrate σ andobtain a quasiconformal map h : C̄ → C̄ ﬁxing 0, ∞, and some b ∈ C∗ with the following
properties:• h → id as→ 0, since the norm of the Beltrami diﬀerential tends to 0.

• h∗ σ0 = σ a.e.
• h is conformal except on the attracting basin of O.
• f := h ◦ g ◦ h−1
 is a rational map which uniformly tends to f as

→ 0.

Remark.
When f (z) = z 2 + c, we may normalize the integrating map h such thath (z)/z → 1 as |z| → ∞ by changing b ∈ C∗ . Then we have f → f of the form f (z) = z 2 +c .

13

Lifting partial conjugacy. Set D := h (D). Then we have a commuting diagramhC̄ − g−1 (D) −−−−→ C̄ − f−1 (D )



f
g 

C̄ − Dh−−−−→

C̄ − D

which actually is:hC̄ − f −1 (D) −−−−→ C̄ − f−1 (D )



f
f

C̄ − Dh−−−−→

C̄ − D .

Since any backward orbits other than the (super)attracting cycles eventually get caught inthis diagram, we naturally have a conjugacy ĥ :ĥAnf −−−−→ Anf



ˆ
fˆ
 f
ĥ

Anf −−−−→ Anf .
More precisely, take any ẑ = (z0 , z−1 , . . .) ∈ Rf . Then there exists an N > 0 such that z−nstays in C̄ − D for all n ≥ N . Setĥ : ẑ = (z0 , z−1 , . . .) → fN (h (zN ), h (z−N −1 ), . . .).
One can easily check that ĥ does not depend on the choice of N and is a conjugacy betweenfˆ and fˆ on their regular leaf spaces. Since h is quasiconformal, the lifted map ĥ is leafwise
quasiconformal by the deﬁnition of the regular leaf spaces. It implies that leaves isomorphic
to C are mapped to leaves isomorphic to C. (Indeed, the map ĥ preserves all types of leaves
in Rf .) Thus the aﬃne parts are preserved by this homeomorphism and we have the diagram
above.


4

Analogy to quasi-Fuchsian deformations

In this section we give an overview of the quotient (hyperbolic) 3-laminations as an analogue ofthe hyperbolic 3-manifolds associated with quasi-Fuchsian groups. In particular, we roughly
explain the relation between the Mandelbrot set and the Bers slice (or the Bers simultaneous
uniformization) in lamination context. This gives an motivation to deﬁne upper and lower
ends for our quotient 3-laminations.

4.1

Quasi-Fuchsian deformation of hyperbolic 3-manifolds

The aim of this paper is to observe topological changes of the quotient 3-lamination as theparameter c of fc moves from one hyperbolic component to another via parabolic g = fσ .
Before proceeding to the investigations, we remark an analogy between quadratic maps andquasi-Fuchsian groups, which is already pointed out in [LM, §6.4, §10]. For the basics ofquasi-Fuchsian groups as π1 of hyperbolic 3-manifolds, see [MT, §4.3.3] and [Mc1, §3.2].
3-Manifolds for once-punctured torus groups.
Let Γ ⊂ PSL(2, R) be a Fuchsiangroup acting on the upper half plane H. Then Γ also stabilizes the lower half plane H− and14the limit set ΛΓ . For example, suppose that S = H/Γ and S = H− /Γ are once-punctured
tori, where S is S with its orientation reversed. In this case the limit set is R ∪ {∞}, andits complement ΩΓ (the region of discontinuity) is given by H  H− . The Fuchsian group Γacts on H3 properly discontinuously and the quotient hyperbolic 3-manifold M = H3 /Γ ishomeomorphic to S × (0, 1). (Topologically it is a bit more natural to consider it a thickened
surface.) Set ∂M := ΩΓ /Γ = S ∪ S and we call it the conformal boundary of M . We say
M := M ∪ ∂M is the Kleinian manifold of M . In this case, we have a natural productstructure M = S × [0, 1]. Let us call S and S the upper and lower ends. It is well known
that the deformation space of Γ is identiﬁed as T (S) × T (S) (where T (S) is the Teichmüller
space of S and isomorphic to H) which corresponds to the deformation of the upper end S
and the lower end S.

Figure 3: Deformation of the hyperbolic 3-manifold associated with a once-punctured torus
group. The heavy curves below show η and ηt .
Quasiconformal
deformation. Now we consider a continuous family of quasiconformalmaps φt : C̄ → C̄ 0<|t|≤1 such that: φ−1 = id; each φt is conformal on H; and the group(Γ−1 = Γ) is a Kleinian group which stabilizes Ht = φt (H) and Ht− = φt (H− )Γt = φt Γφ−1t(=: a quasi-Fuchsian group). Then the quotient Ht /Γt is the same Riemann surface as S,
but S t := Ht− /Γt is quasiconformally deformed. (Such a deformation is realized by the Bers
simultaneous uniformization [Be].) In addition, we may assume that this deformation satisﬁes
the following (see Figure 3) :1. There exists a simple closed geodesic η = η−1 on S = S −1 such that |t| is the hyperbolic
length of the geodesic ηt on S t homotopic to (φt )∗ (η).
2. By letting t increase from −1 to 0, we have an extra cusp at the limit. (Now S 0 is athrice-punctured sphere in this case.)
3. Next as t increases from 0 to 1, we plump the geodesic again but we twist it by a certain
amount of length. More precisely, to have S t , we ﬁx a θ ∈ R and we cut S −t along η−t
and twist by distance tθ to the right, and glue it again.
15

These pinching-and-plumping deformation eventually changes only the complex structure
and the marking of the lower end S. However, it is known that Mt ≈ S × (0, 1) for all t(including t = 0 as a limit) and M t ≈ S × [0, 1] for all t = 0. While this deformation process
the main topological and combinatorial change is observed only in the lower ends.
The Bers slice vs. the Mandelbrot set.
In the deformation space T (S) × T (S),the pinching-and-plumping deformation process above is realized in the Bers slice, which isan embedded image of {[S]} × T (S) ≈ T (S) in a complex Banach space. One may ﬁnd
two distinct paths landing at the same boundary point in the Bers slice that correspond topinching and plumping deformations.
In the following sections we will observe a similar phenomenon as fc moves from one
hyperbolic component to another in the Mandelbrot set. (Not in the same component: Thisis a diﬀerent situation from the Bers slice.) Let us formalize our quadratic setting according
to the quasi-Fuchsian setting.

Figure 4: The Böttcher coordinate Φf gives a conformal conjugacy between f0 |Bf0 and f |Bf .
We start with f0 (z) = z 2 . This map stabilizes the sets Bf0 := C̄ − D̄, Jf0 := {|z| = 1}
and Df0 := D. Let us regard the Mandelbrot set as the deformation space of this map. Set
f (z) := fc (z) = z 2 + c with c in the Mandelbrot set. Then the dynamics of fc gives a naturaldivision of the sphereC̄ = Bf  Df  Jf
where Bf := C̄ − Kf , the basin at inﬁnity, and Df := Kf◦ . Since the ﬁlled Julia set Kf isconnected, Bf is a simply connected domain. Moreover, it is well-known that there exists
a conformal conjugacy (the Böttcher coordinate) Φf : Bf0 → Bf that conjugates f0 |Bf0 and
f |Bf (Figure 4). On the other hand, the structure of Df may change drastically. (It can evenbe empty.)
For simplicity, let c be a parameter in the main cardioid. Then the Julia set Jf is a quasicircle, so Df is a quasidisk. However, c = 0 is the unique parameter with a superattracting
ﬁxed point (except ∞) so there is no quasifoncormal map that conjugates f0 |Df0 and f |Df .
This implies that we cannot regard the main cardioid as the direct analogue of the Bers slices
contained in a quasiconformal deformation space of the holomorphic dynamics on the sphere.
One solution to improve this situation is to consider the dynamics only near the Julia
sets. (This is simply a problem of J-stability.) Another solution we promote is to introduce
the laminations.

4.2

Deformation of the quotient 3-lamination of f0 .

Conformal boundary, upper and lower ends. A little more generally, suppose thatf = fc has an attracting or parabolic cycle. Then Df is non-empty open set containing the16critical point z = 0. Since Bf , Jf , and Df are completely invariant, it is natural to consider
their lift to the aﬃne lamination
Af = Bf  Df  Jf
where Bf := π̂f−1 (Bf ), Df := π̂f−1 (Df ), and Jf := π̂f−1 (Jf ). (Recall that π̂f = πf ◦ ℘. See thediagram in Section 2.)
By the Böttcher coordinate, the dynamics of f : Bf → Bf is conformally conjugate tof0 (z) = z 2 on Bf0 = C − D̄. This conjugacy is naturally lifted to the natural extension. By
restricting it on the aﬃne part, or equivalently, on the aﬃne lamination, we have a conformal
conjugacy between the invariant sublaminations Bf0 of Af0 and Bf of Af .
The structure of Bf0 is described as follows: (See also [LM, §11] and [S].) Under theidentiﬁcation of T = R/Z as ∂D = Jf0 , the angle doubling δ : T → T, θ → 2θ is considered
as a covering of degree two on the unit circle. Now the Julia set upstairs Jf0 is identiﬁed
as the inverse limit T̂ of the dynamics δ : T → T, which is a one-dimensional lamination.
(See §6.1.) Each leaf of T̂ = Jf0 is isomorphic to R with the natural aﬃne structure comingfrom T. We can attach H to each leaf of T̂, and this is exactly what Bf0  Jf0 is. The actionfˆ0 : Bf0 → Bf0 is leafwise isomorphic and properly discontinuous. The quotient Bf0 /fˆ0 iscalled Sullivan’s solenoidal Riemann surface lamination S0 , which is compact and consists ofleaves isomorphic to H or annuli.
By conformal equivalence, the Riemann surface lamination Bf /fˆ is equivalent to S0 for
any f . In addition, by the argument in [LM, §§3.5] or [S], we have:
Proposition 4.1 (Conformal boundary) Suppose f = fc has an attracting or parabolic
cycle. Then both Bf and Df are Riemann surface sublaminations of Af and that the actionsof fˆ on Bf and Df are properly discontinuous. Thus the quotients Bf /fˆ and Df /fˆ are again
Riemann surface laminations. In particular, Bf /fˆ is conformally the same as S0 := Bf0 /fˆ0 .
Hence Bf /fˆ and Df /fˆ form the two ends of Mf corresponding to the ends S and St of Mt
above. We say S0 = Bf /fˆ the upper end, and Sf := Df /fˆ the lower end. Now the union
S0 ∪ Sf is ∂Mf = Ff /fˆ, the conformal boundary of Mf (Section 2). Recall that we say
Mf = ∂Mf ∪ Mf is the Kleinian lamination associated with f .
Example: c = 0. Since f0 has a symmetric dynamics with respect to Jf0 (the unit circle),
Df0 is the mirror image of Bf0 . Thus the lower end Sf0 = Df0 /fˆ0 is the mirror image S0 of
S0 , so we have a natural product structure Mf0 ≈ S0 × [0, 1] (Figure 5).

Figure 5: The Kleinian lamination of f0 .
Deformation of the Kleinian lamination Mf0 . Now let us consider a deformation Mf
of Mf0 within the main cardioid. As an analogue to quasi-Fuchsian groups, we have:17

Theorem 4.2 (Deformation in the main cardioid) For any f = fc with c in the main
cardioid, the laminations Af , Bf , Df , Hf , and Mf are quasiconformally or quasi-isometricallyequivalent to Af0 , Bf0 , Df0 , Hf0 , and Mf0 respectively.
In particular, the Kleinian lamination Mf = Mf ∪ ∂Mf is homeomorphic to the product
S0 × [0, 1] ≈ Sf × [0, 1].
This theorem emphasize the similarity between the Bers slice and the main cardioid of theMandelbrot set. Note that the existence of the product structure in Mf is already shown in[LM, §6.4]. Here we give an alternative proof:
Proof. The ﬁrst paragraph directly comes from Corollary 3.4. Then last sentence follows
from the product structure Mf0 ≈ S0 × [0, 1].
Remarks.
• In Section 10 we will show that the quadratic Kleinian lamination Mf has a productstructure if and only if c of f = fc is in the main cardioid (Theorem 10.6).
• The theorem above is true even if we replace “f = fc in the main cardioid” by “f in thehyperbolic component of f0 (z) = z d in the space of rational function of degree d ≥ 2”.
For this case, the Riemann surface lamination Df is also dealt with in [Mc2, §10] as ananalogue of the foliated unit tangent bundle over a Riemann surface.
The following sections are devoted to the thorough study of the degeneration process of
the aﬃne, H3 -, and quotient 3-laminations and their conformal boundaries. According to thedeformation process in the quasi-Fuchsian example above, the interior of the quotient 3-space
would be topologically preserved, but its conformal boundary (the lower end) would change.
We will also study the bifurcation of the degenerated laminations, which would not happen
in the quasi-Fuchsian deformation space.

5

Degeneration pairs and Tessellation

From this section we return to the setting of Part I [Ka3] and investigate degenerations andbifurcations of quadratic maps in terms of the Lyubich-Minsky laminations.
Abstract of this section. We summerize the notation and results in Part I as follows:• We formalize hyperbolic-to-parabolic degeneration of a quadratic map in the Mandelbrot set in terms of a degenetion pair (f → g) with hyperbolic f and parabolic g. We
divide the degeneration pairs in two types: Case (a) and Case (b).
• For hyperbolic f above, we deﬁne the degenerating arc system If which corresponds to
the grand orbit Ig of the parabolic cycle of g.
• We deﬁne the tessellations Tess(f ) and Tess(g) for (f → g). Tessellations are tilings ofthe interiors of the ﬁlled Julia sets. Each tile is of the form T = T (θ, m, ∗) parameterized
by angle θ, level m, and signature ∗ = ± (Theorem 5.3).
• The boundary of each tile is the union of three or four analytic arcs (“edges”). We
classify them into equipotential, critical, and degenerating edges.
• By using tessellations, we have a pinching semiconjugacy h : C̄ → C̄ from f to g thatjust pinches If to Ig (Theorem 5.5).
18

The objects above will play key roles in the following sections. This section is a little more
expository than the original [Ka3], where detailed deﬁnitions and proofs are given.

5.1

Degeneration pairs: Case (a) vs. Case (b) ([Ka3, §2])

Degeneration pair. Let X be a hyperbolic component of the Mandelbrot set. By Douady
and Hubbard’s uniformization theorem [Mi, Theorem 6.5], there exists a conformal map λX
from D onto X that parameterize the multiplier of the attracting cycle Of of f = fc forc ∈ X. Moreover, the map λX has a unique homeomorphic extension λX : D̄ → X̄ such thatλX (e2πip/q ) is a parabolic parameter for all p, q ∈ N. A degeneration pair (f → g) is a pairof hyperbolic f = fc and parabolic g = fσ where (c, σ) = (λX (re2πip/q ), λX (e2πip/q )) for some
0 < r < 1 and coprime p, q ∈ N. By letting r → 1, we have a convergence f → g uniformly
on C̄.
Case (a) and Case (b). Let us introduce more notation that will be used throughout thispaper. For a degeneration pair (f → g), let Of := {α1 , . . . , αl } (taking subscripts modulol) be the attracting cycle of f and Og := {β1 , . . . , βl } (taking subscripts modulo l ) be the parabolic cycle of g. Let re2πip/q and e2πip /q denote the multipliers of Of and Og withcoprime p and q  in N. (Then Og is a parabolic cycle with q  repelling petals.) Then we have
Proposition 5.1 ([Ka3, Proposition 2.1]) Any degeneration pair (f → g) satisﬁes either
Case (a):q = q  and l = l ; orCase (b):q = 1 < q  and l = l q  .

l.
For both cases, we have lq = l q  =: ¯For example, if f is in the main cardioid of the Mandelbrot set and g is the root of asatellite limb, then (f → g) is Case (a). Indeed, any Case (a) degeneration pair is either
a tuned copy of such (f → g); or g corresponds to a cusped point of a small copy of theMandelbrot set. (See Figure 6.) It would be helpful to remember that Case (a) happens iﬀ
the attracting points of f and the parabolic points of g have one-to-one correspondence.
Examples: The Cauliﬂowers. The simplest example is given when g = f1/4 and f = fc
with 0 < c < 1/4. In this case we have p = p = 1, q = q  = 1, and l = l = 1 thus Case (a).
We call such a degeneration pair the Cauliﬂowers.
Examples: The Rabbits. By Douady’s rabbit we mean the quadratic map fcrab whichis the center of the 1/3-limb of the Mandelbrot set. (In this paper we often identify thequadratic map fc (z) = z 2 + c and the parameter c.) By the Rabbits we mean a deformation
process from f0 (z) = z 2 to fcrab (z) = z 2 + crab that can be represented by two degeneration
pairs (f1 → g) and (f2 → g) as follows: Let g = fσ be the root of the 1/3-limb. Then we cantake a Case (a) degeneration pair (f1 → g) with f1 = fc1 in the main cardioid and a Case(b) degeneration pair (f2 → g) with f2 = fc2 in the same hyperbolic component as Douady’s
rabbit. (See also Figure 1 of Part I [Ka3].)
For the Case (a) Rabbits (f1 → g), we have p = p = 1, q = q  = 3, and l = l = 1. For theCase (b) Rabbits (f2 → g), we also have p = p = 1 but q = 1 < q  = 3 and l = 3 > l = 1. (SeeFigures 7 and 8 below.) This represents a standard example of degeneration and bifurcation
processes. Note that lq = l q  = 3 is invariant under this example.
Examples: The Airplanes.
The only real quadratic map fcair (z) = z 2 + cair withsuperattracting cycle of period three is called the Airplane. This parameter is the center of19

Figure 6: Suppose that this is a part of a primitive copy of the Mandelbrot set. (For example,
the cuspidal point on the boundary of the shadowed cardioid can be c = 1/4, c = −7/4, etc.)
The blue dotted arrows and black arrows show the Case (a) and Case (b) degeneration pairs
respectively.
a primitive copy of the Mandelbrot set. We say a degeneration pair (f → g) = (fc → fσ ) is
the Airplanes if cair < c < −7/4 and σ = −7/4. In this case we have p = p = 1, q = q  = 1,
and l = l = 3 thus Case (a).

5.2

Degenerating arc system ([Ka3, §2])

Degenerating arc system. For a degeneration pair (f → g) with r ≈ 1, the parabolic
cycle Og is approximated by an attracting or repelling cycle Of with the same period l and 
multiplier λ ≈ e2πip /q . Let α1 be the point in Of with α1 → β1 as r → 1.
In Case (a), the cycle Of is attracting (thus Of = Of ) and there are q  symmetrically
arrayed repelling periodic points around α1 = α1 . Then there exits an f l -invariant star-likegraph I(α1 ) that joins α1 and the repelling periodic points by q  arcs.
In Case (b), the cycle Of is repelling and there are q  = l/l symmetrically arrayed
attracting periodic points around α1 . Then there exits an f l -invariant star-like graph I(α1 )that joins α1 and the attracting periodic points by q  arcs.

Figure 7: Invariant graphs for the Rabbits. The arrows show the actions of f13 , g3 , and f23 .
In both cases, the choice of I(α1 ) has an ambiguity. Here we make a speciﬁc choice according to [Ka3, Lemma 2.3]: The arcs in I(α1 ) and the critical orbit are lying on symmetrically20arrayed radial rays in the Königs coordinate. We deﬁne the degenerating arc system If by
f −n (I(α1 )).

If :=
n≥0

As its (expected) limit as f → g, we deﬁne Ig by
g−n ({β1 }).

Ig :=
n≥0

Note that If and Ig are forward and backward invariant sets. For later use, let αf be the setof all points which eventually land on Of . For ζ ∈ αf , let I(ζ) denote the path-connectedcomponent of If (simply, If -component) that contains ζ. In Case (b), each If -component
contains q  points in αf .
Example: The Cauliﬂowers. Let α1 and γ1 be the attracting and repelling ﬁxed pointsof f respectively. Let β1 = 1/2 be the parabolic ﬁxed point of g. In this case we may
regard α1 as α1 and the invariant arc I(α1 ) is just the interval [α1 , γ1 ] on the real axis. The
degenerating arc system If is the union of Jordan arcs that eventually land on the interval[α1 , γ1 ], and Ig is the union of points that eventually land on β1 . (See Figure 9, the thicksegments.) The case of the Airplanes is similar.
Example: The Rabbits. The invariant arc I(α1 ) with nearby dynamics for the Rabbits
f1 and f2 are drawn in the left and right hand sides of Figure 7 respectively. By taking its
preimages, we have If1 and If2 as in Figure 8.

Figure 8: The left panel shows the ﬁlled Julia set of f1 with their degenerating arc systemroughly drawn in. The right panel shows the same for f2 . Attracting cycles are shown inheavy dots. Degenerating arcs with types (§5.3) {1/7, 2/7/4/7} and {1/28, 23/28, 25/28} areemphasized.

5.3

Tessellations associated with degeneration pairs ([Ka3, §3])

Types. Let Jf and Kf denote the Julia set and the ﬁlled Julia set of f . For any degeneration
pair (f → g), we have If ⊂ Kf and Ig ⊂ Jg . It is known that both Jf and Jg are locallyconnected so all the external rays land at these Julia sets. Let Rf (θ) denote the external rayof angle θ and γf (θ) denote its landing point in the Julia set.
21

For a point z ∈ Jf or Jg , its type Θ(z) is the set of angles of external rays that land onz. We abuse this term for more general subsets in the ﬁlled Julia sets Kf and Kg . For any
subset E of the ﬁlled Julia set, its type Θ(E) is the set of angles of external rays that landon points in E. Let Θf and Θg denote Θ(If ) and Θ(Ig ) respectively.
Example: The Cauliﬂowers. We again consider the case of the Cauliﬂowers. Since therepelling ﬁxed point γ1 is γf (0) and the parabolic ﬁxed point β1 is γg (0), both Θf and Θgconsist of the angles that eventually land on 0 by the angle doubling map δ : θ → 2θ. HenceΘf = Θg and they are dense subsets of T = R/Z.
Example: The Rabbits. For (f1 → g), the set Θf1 consists of the angles that eventually
land on {1/7, 2/7, 4/7} by angle doubling. One can also show that Θf1 = Θg and they aredense subsets of T = R/Z. We also have Θf2 = Θg for (f2 → g).
Example: The Airplanes. We also have Θf = Θg and this set consists of angles thateventually land on {1/7, 2/7, 3/7, 4/7, 5/7, 6/7}. See [Ka3, Figure 3].
In general, we have:
Proposition 5.2 ([Ka3, Proposition 2.5]) For any degeneration pair (f → g), the set
Θf = Θ(If ) coincides with Θg = Θ(Ig ).
We denote this set Θf = Θg by Θ = Θ(f →g) .
Tessellation. The dynamics on/outside the Julia set is organized by the external rays andtheir landing points, satisfying
f (Rf (θ)) = Rf (2θ)

and f (γf (θ)) = γf (2θ).

Tessellation is a method that organizes the dynamics inside the Julia set. Each tile is anelement like an external ray outside:
Theorem 5.3 (Tessellation [Ka3, Theorem 1.1]) Let (f → g) be a degeneration pair.
There exist families Tess(f ) and Tess(g) of simply connected sets with the following properties:(1) Each element of Tess(f ) is called a tile and identiﬁed by an angle θ in Θ = Θ(f →g) , a
level m in Z, and a signature ∗ = + or −.
(2) Let Tf (θ, m, ∗) be such a tile in Tess(f ). Then f (Tf (θ, m, ∗)) = Tf (2θ, m + 1, ∗).
(3) The interiors of tiles in Tess(f ) are disjoint topological disks. Tiles with the same
signature are univalently mapped each other by a branch of f −i ◦ f j for some i, j > 0;
(4) Let Πf (θ, ∗) denote the union of tiles with angle θ and signature ∗. Then its interiorΠf (θ, ∗)◦ is also a topological disk and its boundary contains the landing point γf (θ) of
Rf (θ). In particular, f (Πf (θ, ∗)) = Πf (2θ, ∗).
The properties above hold if we replace f by g. Moreover:(5) The union of tiles in Tess(f ) is Kf◦ − If . On the other hand, the union of tiles inTess(g) is Kg◦ .
(6) The boundaries of Tf (θ, m, ∗) and Tf (θ  , m , ∗ ) in Kf◦ − If intersect iﬀ so do the boundaries of Tg (θ, m, ∗) and Tg (θ  , m , ∗ ) in Kg◦ .

22

Figure 9: Tessellations for the Cauliﬂowers and their construction. The blue curves withnumbers are external rays with its angles. The numbers in green indicate the levels of tiles.
The black thick curves show the degenerating arcs. The orange curves correspond to thecritical edges of the tiles. See Figure 11.

23

One can easily understand the idea of tessellation by looking at the following ﬁgures(Figures 9 and 10, [Ka3, Figure 2], etc.), but the construction is not that easy. Here weroughly explain the construction and the combinatorics of the addresses by showing somestandard examples.
Example: The Cauliﬂowers. Their tessellations are shown in Figure 9. The tiles are
based on fundamental regions of the model dynamics given by F : W → rW + 1 (0 < r < 1)
and G : W → W + 1 on C. The map F has an attracting ﬁxed point a = 1/(1 − r), andan invariant interval I = [a, ∞) ⊂ R. An important fact is that F : C − I → C − I is
topologically conjugate to G : C → C. For example, we can ﬁnd a conjugacy that sends the“semi-fundamental region”A(m, ±) := W ∈ C − I : r m+1 a ≤ |W − a| ≤ r m a, ± Im W ≥ 0
toC(m, ±) := {W ∈ C : m ≤ Re W ≤ m + 1, ± Im W ≥ 0}
for any m ∈ Z and any choice of + or −.
We can relate these models and the original dynamics as follows. Now we may regard r
as the multiplier of the attracting cycle α1 . The dynamics near α1 is conformally conjugate
to w → rw by a Königs coordinate, but we take an extra conjugacy by an aﬃne map so that
it is conformally conjugate to F near a. In fact, there exists a modiﬁed Königs coordinate
Φf : Kf◦ → C such that: Φf (0) = 0; Φf is a branched covering semiconjugating f |Kf◦ and
F on C; the degenerating arc system If is mapped onto the interval I; and we can take+countably many univalent branches of Φ−1
f over the upper and lower half-planes H = H and
−
H .
±
±
Let us choose speciﬁc branches Ψ±θ over H so that the images of H contain γf (θ) ∈If ⊂ Jf in their boundaries. These images actually give the interiors of the panels Πf (θ, +)or Πf (θ, −). (Thus θ must be in Θ.)±
One can easily see that we have a homeomorphic extension Ψ±θ : H → C. Now the tileof address (θ, m, ±) in Tess(f ) is deﬁned by
Tf (θ, m, ±) := Ψ±θ (A(m, ±)).
Correspondingly, we have the normalized Fatou coordinate Φg : Kg◦ → C so that Φg (0) = 0
and Φg is a branched covering semiconjugating g|Kg◦ and G(W ) = W + 1 on C. In particular,
over H± . To deﬁne the tile
we can also take countably many univalent branches of Φ−1g±±Tg (θ, m, ±), we take the branches Ψθ over H so that the images of H± contain γg (θ) ∈ Ig ⊂±Jg in their closures. Since we also have a homeomorphic extension Ψ±θ : H → C, we deﬁne
the tile of address (θ, m, ±) in Tess(g) byTg (θ, m, ±) := Ψ±θ (C(m, ±)).
Example: The Rabbits. Figure 10 shows tessellations Tess(f1 ) and Tess(f2 ). It is diﬃcult
to draw a nice picture of the tessellation for g because a terrible moire pattern appears. But
one can imagine it by pinching If1 or If2 . The construction of these tessellations are again
based on the modiﬁed Königs coordinates and the normalized Fatou coordinates. (Here weskip the details.)
By looking at Figure 10 carefully, one would ﬁnd that the tessellations for g associatedwith (f1 → g) and (f2 → g) would be diﬀerent. In fact, the former is a subdivided tessellation

24

Figure 10: Tessellations for the Rabbits f1 and f2 . (cf. The Basilicas [Ka3, Figure 7]) By
Theorem 5.5, the tessellations for g associated with (f1 → g) and (f2 → g) are given by justpinching If1 and If2 .

25

of the latter. The diﬀerence comes from the diﬀerence of p, q and l for f1 and f2 . (We willdeal with this diﬀerence in Section 10.)
Edges of tiles ([Ka3, §§3.2 - 3.3]). For the degeneration pair (f → g), every tile T in
Tess(f ) (resp. Tess(g)) is quadrilateral (resp. trilateral) with piecewise analytic boundaryarcs, which we call edges. It is useful to call these edges as follows (see Figure 11):• Degenerating edges: For each tile T in Tess(f ), its boundary ∂T contains an arc
shared with If . We call such an arc the degenerating edge of T . Note that the degenerating edge is not contained in T itself. On the other hand, for each tile T  in
Tess(g), the part corresponding to the degenerating edge in its boundary ∂T  is thevertex contained in Ig . Note also that this point is not contained in T  itself.
• Equipotential edges: For each ∂T and ∂T  there are two edges lying on the equipotential curves with respect to the linearizing (Königs or Fatou) coordinate. When we
make panels these edges are glued. We call them the equipotential edges or the circularedges.
• Critical edges: For each ∂T and ∂T  there is one edge left. Such an edge is containedin an arc joining two points in the grand orbit of the critical point z = 0. We call sucharcs the critical edges.

Figure 11: Naming the edges of tiles.
The combinatorics of edges will play an important role in Sections 9 and 10. Here we givea reﬁnement of Theorem 5.3(6):Proposition 5.4 ([Ka3, Proposition 3.2]) The edges of tiles have the following properties:• The equipotential (resp. critical) edges of Tf (θ, m, ∗) and Tf (θ  , m , ∗ ) are shared iﬀ so
do the equipotential (resp. critical) edges of Tg (θ, m, ∗) and Tg (θ  , m , ∗ ).
• If the degenerating edges of Tf (θ, m, ∗) and Tf (θ  , m , ∗ ) are shared, then Tg (θ, m, ∗)and Tg (θ  , m , ∗ ) share their boundary points in Ig . That is, γg (θ) = γg (θ  ).
One can easily observe these facts in Figures 9 and 10.

5.4

Pinching semiconjugacy ([Ka3, §4 - 5])

As an application of tessellation, we can show that there exists a pinching semiconjugacy
from f to g for the degeneration pair (f → g):
26

Theorem 5.5 (Theorem 1.2 of [Ka3]) Let (f → g) be a degeneration pair. There exists
a semiconjugacy h : C̄ → C̄ from f to g such that:(1) h only pinches each If -component to the point in Ig with the same type.
(2) h sends all possible Tf (θ, m, ∗) to Tg (θ, m, ∗), Rf (θ) to Rg (θ), and γf (θ) to γg (θ).
(3) h tends to the identity as f tends to g.
So we have a precise picture for the degeneration of the dynamics. An interesting fact is that
this theorem is proved without using quasiconformal deformation.
In the following sections we will investigate degeneration of the laminations by using thetessellations and this semiconjugacy.

6

Degeneration of the aﬃne laminations

In this section we consider degeneration of the natural extentions, aﬃne parts, and aﬃnelaminations for (f → g).
We ﬁrst lift all the objects deﬁned in the previous section to the natural extensions. For
example, we will consider the external rays, degenerating arcs, and tessellations upstairs.
(Such objects are nicely parameterized by the natural extention T̂ of the angle doubling map
δ : T → T.)
Next in Theorem 6.1, we will construct a natural lift ĥ : Nf → Ng of the pinching
semiconjugacy h : Ĉ → Ĉ given by Theorem 5.5. This lifted pinching semiconjugacy ĥ will
give a precise picture of the degeneration of Nf to Ng . By checking the action of ĥ on theaﬃne part Anf , we also describe the degeneration of Anf to Ang . Then we will conclude thatas f → g the leafwise topology of Anf are preserved except on some speciﬁc leaves, called theprincipal leaves (Theorem 6.6). The descriptions on the aﬃne parts are directly translated
to the aﬃne laminations Af and Ag by Corollary 2.3.
The results in this section are closely related to the backward Julia sets of the Hénonmappings. See Appendix A for more details.

6.1

Semiconjugacies in the natural extension

First we describe the degeneration of the natural extensions associated with (f → g). Let
us start with lifting the objects in C̄ (“downstairs”) to the natural extensions Nf and Ng(“upstairs”).
Natural extension of the angle doubling. To parameterize the objects we will deal
with, let us consider the natural extension T̂ of the angle doubling map δ : T → T, θ → 2θ.
That is,T̂ := {θ̂ = (θ0 , θ−1 , . . .) : θ0 ∈ T, 2θ−n−1 = θ−n } ⊂ T × T × · · ·with a natural homeomorphic action δ̂ : T̂ → T̂ given byδ̂(θ̂) := (2θ0 , θ0 , θ−1 , . . .).
We also call the elements of T̂ angles. Let Θ̂ = Θ̂(f →g) denote the invariant lift of Θ = Θ(f →g)
in T̂. This subset of angles upstairs will play exactly the same role as Θ downstairs.
External rays upstairs. External rays outside the Julia sets are naturally lifted to thenatural extensions. For θ̂ = (θ−n )n≥0 ∈ T̂, we say ẑ = (z−n )n≥0 ∈ Rf (θ̂) if z−n ∈ Rf (θ−n )27for all n ≥ 0. That is, the external ray Rf (θ̂) is the backward orbits (or history) along thesequence
Rf (θ0 ) ← Rf (θ−1 ) ← Rf (θ−2 ) ← · · · .
Note that any ẑ ∈ Rf (θ̂) is regular, thus Rf (θ̂) ⊂ Rf (= Anf by Proposition 2.1). Since
every external ray Rf (θ−n ) lands on Jf , it is reasonable to denote the backward orbit(γf (θ0 ), γf (θ−1 ), . . .) ∈ πf−1 (Jf ) by γf (θ̂). Hence the Julia set upstairs Jfn := πf−1 (Jf ) isparameterized by T̂. Indeed, one can check that γf : T̂ → Jfn is a continuous map. We deﬁne
Rg (θ̂) and γg (θ̂) in the same way.
Types upstairs. Let ẑ = (z0 , z−1 , . . .) ∈ Nf be a point in Jfn . We deﬁne the type Θ(ẑ) ofẑ by the set of angles of the form θ̂ = (θ−n )n≥0 ∈ T̂ satisfying θ−n ∈ Θ(z−n ) for all n ≥ 0.
One can easily check that δ̂(Θ(ẑ)) = Θ(fˆ(ẑ)). (Note that the critical orbit of f is boundeddistance away from Jf . Thus card(Θ(ẑ)) is ﬁnite for each backward history ẑ in Jf . See[Ka3, §2].) Types for backward orbits in Jg are also deﬁned in the same way.
Degenerating arcs upstairs. Let us set α̂f = π −1 (αf ), Iˆf := π −1 (If ) and Iˆg := πg−1 (Ig ).
ff

For ζ̂ = (ζ0 , ζ−1 , . . .) ∈ α̂f , we deﬁne I(ζ̂) by the set of backward orbits ẑ = (z−n )n≥0 withz−n ∈ I(ζ−n ) for all n ≥ 0. Equivalently, the set I(ζ̂) is the path-connected component ofIˆf (simply, Iˆf -component) containing ζ̂ in Nf . The type Θ(I(ζ̂)) of I(ζ̂) is the ﬁnite set ofangles of the form θ̂ = (θ−n )n≥0 ∈ Θ̂ satisfying θ−n ∈ Θ(I(ζ−n )) for all n ≥ 0.
Tiles upstairs.
The tessellations are naturally lifted to the natural extensions. For
θ̂ = (θ−n )n≥0 ∈ Θ̂ and ẑ = (z−n )n≥0 ∈ Nf , we say ẑ ∈ Tf (θ̂, m, ∗) if Tf (θ0 , m, ∗) ∈ Tess(f )and z−n ∈ Tf (θ−n , m − n, ∗) for all n ≥ 0. Since tiles in Tess(f ) are homeomorphically
mapped each other, the tile Tf (θ̂, m, ∗) is contained in Rf = Anf and the projection πf on
Tf (θ̂, m, ∗) is a homeomorphism onto Tf (θ0 , m, ∗). We gather all such tiles and denote it byTessn (f ). One can easily check that Tessn (f ) actually tessellates πf−1 (Kf◦ − If ). For g, wedeﬁne tiles and tessellation upstairs in the same way.
Pinching semiconjugacy upstairs. Now we have a natural lift of Theorem 5.5:
Theorem 6.1 (Natural extensions) For degeneration pair (f → g), there exists a semiconjugacy ĥ = ĥn : Nf → Ng from fˆ to ĝ with the following properties:(1) ĥ only pinches each Iˆf -component to the point in Iˆg with the same type.
(2) ĥ sends all possible Tf (θ̂, m, ∗) to Tg (θ̂, m, ∗), Rf (θ̂) to Rg (θ̂), and γf (θ̂) to γg (θ̂).
ˆ ĥ(ẑ), ẑ) tends to 0(3) ĥ tends to the identity as f tends to g. More precisely, the distance d(ˆuniformly for any ẑ ∈ Nf where d(·, ·) is a distance function on C̄ × C̄ × · · · .
We will use this lifted semiconjugacy to investigate the degenerations of the laminations.
Proof. For ẑ = (z0 , z−1 , . . .) ∈ Nf , setĥ(ẑ) := (h(z0 ), h(z−1 ), . . .) ∈ Ng .
Since h is a semiconjugacy from f to g, one can easily check that ĥ is surjective, continuous,and satisﬁes ĥ ◦ fˆ = ĝ ◦ ĥ. Thus ĥ is a semiconjugacy from fˆ to ĝ on their respective naturalextensions.
Now properties (1)–(3) are almost straightforward. To show (3), one can choose theˆ ζ̂) :=
distance as follows: For ẑ = (z−n )n≥0 and ζ̂ = (ζ−n )n≥0 in C̄ × C̄ × · · · , set d(ẑ,
28



n
ˆ
n≥0 dC̄ (z−n , ζ−n )/2 , where dC̄ (·, ·) is the spherical distance on C̄. Then d(·, ·) determines

the same topology as the natural topology of C̄ × C̄ × · · · .

Remark. Note that property (3) of the theorem implies Nf → Ng as f → g in the Hausdorﬀ
topology of non-empty compact subsets in C̄ × C̄ × · · · . Indeed, for any uniform convergenceof rational maps fn → f , it is always true that Nfn → Nf in the Hausdorﬀ topology.

6.2

Principal leaves

Next we consider degeneration of the aﬃne part Anf to Ang . To state the main theorem(Theorem 6.6), we need to deﬁne some key objects: The irregular cycles Ôf and Ôg , principal
leaves Λf and Λg , and the cyclic degenerating arcs I(Ôf ).
Irregular cycles. For the attracting cycle Of = {α1 , . . . , αl } with f (αj ) = αj+1 (taking
subscripts modulo l), we deﬁne its cyclic lift Ôf = {α̂1 , . . . , α̂l } byα̂j := (αj , αj−1 , αj−2 , . . .) ∈ Nf
for each j modulo l. For the parabolic cycle Og = {β1 , . . . , βl } with g(βj  ) = βj  +1 , wedeﬁne Ôg = {β̂1 , . . . , β̂l } in the same way by taking subscripts modulo l . In the case ofq = 1 < q  (“Case (b)” in Section 5), the pinching semiconjugacy h gives a natural one-to-one
correspondence between l attracting points in Of and l = q  l attracting directions associatedwith Og . So we assume that αj  , αj  +l , . . . , αj  +(q −1)l are pinched to βj  by h; or equivalently,
I(αj  ) = I(αj  +l ) = · · · = I(αj  +(q −1)l )(6.1)

and this If -component has the same type as βj  . Note that we always have the relationf (I(αj  )) = I(αj  +1 ), which corresponds to g(βj  ) = βj  +1 by the semiconjugacy h.
Remark. In both Case (a) and Case (b), it is economical to use the subscript j  modulol . This implies that the properties of g are dominant when we consider degeneration of theaﬃne parts.
Examples. The Cauliﬂowers (f → g) have the irregular cycles Ôf and Ôg which are justthe lifted attracting and parabolic ﬁxed points {α̂1 } and {β̂1 }. The same is true for theCase (a) Rabbits (f1 → g), but the Case (b) Rabbits (f2 → g) have more irregular points
in Ôf2 : We have Ôf2 = {α̂1 , α̂2 , α̂3 }, the cyclic lift of the attracting cycle {α1 , α2 , α3 } withπf (α̂j ) = αj . Note that the equality (6.1) corresponds to I(α1 ) = I(α2 ) = I(α3 ).
The aﬃne parts as the aﬃne laminations.
fairly easy to specify the aﬃne parts Anf and Ang :For our degeneration pair (f → g), it is

Proposition 6.2 The aﬃne parts of (f → g) are given byˆ and Ang = Ng − Ôg ∪ {∞}.
ˆAnf = Nf − Ôf ∪ {∞}Moreover, Anf and Ang are Riemann surface laminations with leaves isomorphic to C, which
are conformally equivalent to the aﬃne laminations Af and Ag in the universal setting.
Proof. For f and g, there is a common irregular point ∞ˆ = (∞, ∞, . . .) in their naturalextensions. By Proposition 2.1, we have the equalities as above.
Recall that the aﬃne parts Anf and Ang are the unions of all leaves isomorphic to C in theregular leaf spaces. Moreover, we may identify Anf and Ang as the aﬃne laminations Af and29n
Ag by Corollary 2.3. More precisely, the map ℘|Af = ι−1
f : Af → Af gives the conformal
conjugacy between fˆa and fˆn |An thus Af is conformally equivalent to An . The same holdsff

for g.



By using this identiﬁcation, we will translate the following investigation on the aﬃne parts
to the aﬃne laminations.
Principal leaves. Let us introduce the important notion of principal leaves where we willobserve the main diﬀerence caused by the hyperbolic-to-parabolic degeneration.
For each j  modulo l , we deﬁne the following subsets in the aﬃne parts:Λ(α̂j  ) := ẑ = (z0 , z−1 , . . .) ∈ Anf : {z−nl }n≥0 accumulates on I(αj  ) ; andΛ(β̂j  ) := ẑ = (z0 , z−1 , . . .) ∈ Ang : {z−nl }n≥0 accumulates on βj  .
Note that they do not contain the irregular points α̂j  and β̂j  themselves. In addition, we
can replace “I(αj  )” in the the deﬁnition of Λ(α̂j  ) by “the repelling points in I(αj  )” .
Then the dynamics downstairs implies:
Proposition 6.3 (Principal leaves) The objects deﬁned above have the following properties:(1) For each j  modulo l , fˆ(Λ(α̂j  )) = Λ(α̂j  +1 ) and ĝ(Λ(β̂j  )) = Λ(β̂j  +1 ).
(2) Case (a): If q = q  and l = l , then for each j  modulo l , the set Λ(α̂j  ) consists of q leaves cyclic under fˆl , and each of such leaves consists of the backward orbits converging
to one of the q  repelling periodic points in I(αj  ). Similarly, the set Λ(β̂j  ) also consistsof q  leaves cyclic under ĝl , and each of such leaves consists of the backward orbits
converging to one of the q  repelling petals of βj  .
(3) Case (b): If q = 1 < q  and l = l q  , then for each j  modulo l , the set Λ(α̂j  ) is just one
leaf invariant under fˆl which consists of the backward orbits converging to the central
repelling periodic point of I(αj  ). Moreover, we haveΛ(α̂j  ) = Λ(α̂j  +l ) = · · · = Λ(α̂j  +(q −1)l ).
On the other hand, each Λ(β̂j  ) consists of q  leaves described as above.
(4) In both cases above, we have lq = l q  =: ¯l. Every leaf in Λf (α̂j  ) or Λg (β̂j  ) is invariant
under fˆl̄ or ĝl̄ respectively.
(5) There exists a unique λ with |λ | > 1 such that for any j  modulo l and any leaf L ofΛf (α̂j  ), the map fˆl̄ : L → L is conformally conjugate to the hyperbolic action w → λ w
on C. Similarly, for any j  modulo l and any leaf L of Λg (β̂j  ), the map ĝl̄ : L → Lis conformally conjugate to the parabolic action w → w + 1. Moreover, the multiplier λtends to 1 as f → g.
The proof is almost straightforward and left to the reader. (The last property is given by
the existence of the linearizing coordinates near repelling cycles and petals.) To simplify thenotation, we set
l
l


Λ(α̂j  ) and Λg :=Λ(β̂j  ).
Λf :=j  =1j  =130

We call Λf and Λg (and the leaves in these sets) the principal leaves, where we will observea signiﬁcant degeneration associated with (f → g). The following examples are helpful:
Example: The Cauliﬂowers.
For the Cauliﬂowers (f → g), the principal leaves Λfand Λg consist of backward orbits converging to the repelling and parabolic ﬁxed points
respectively. They are invariant leaves of fˆ and ĝ with hyperbolic and parabolic dynamics
respectively (Figure 12). Essentially the uniformizations of these principal leaves are given
by uniformizing the quotient torus and the cylinder of the dynamics downstairs. See [Ka1]and [LM, §4.2].

Figure 12: The arrows above show the local dynamics near α1 and β1 for the Cauliﬂowers(f → g). The two panels below illustrate the principal leaves. The black heavy line (on theleft panel) is I(Ôf ), and the blue lines are the external rays of angle θ̂ = (0, 0, . . .). The green
curves show the fundamental regions.
Example: The Rabbits. For f1 (= Case (a)) and g , both Λf1 and Λg consist of q = q  = 3
leaves which are permuted by the actions of fˆ1 and ĝ. On the other hand, for f2 (= Case(b)), Λf2 is just one (= q) leaf which is invariant under the action of fˆ2 . However, we haveobvious correspondence between the dynamics on Λf1 , Λg , and Λf2 . (See Figure 13.)Cyclic degenerating arcs. As the parabolic cycles Og and Ôg are important elements ofthe dynamics of g and ĝ, the corresponding objects
I(Of ) := h−1 (Og ) and I(Ôf ) := ĥ−1 (Ôg )play very important roles in the dynamics of f and fˆ. We call them the cyclic degeneratingarcs of f . Note that I(Ôf ) upstairs contains irregular points Ôf .
By Theorem 6.1 and Proposition 6.3, it is not diﬃcult to check the following:31

Figure 13: The principal leaves of the Rabbits f1 , g, and f2 (below). The three colors(blue, green, pink) indicate the three repelling directions of the parabolic cycle β1 , and thecorresponding dynamics for f1 and f2 . The black heavy lines indicate part of the cyclic degenerating arcs I(Ôf1 ) and I(Ôf2 ). The arrows show the actions of f13 , g3 , and f23 downstairs.
Proposition 6.4 (Cyclic degenerating arcs) The cyclic degeneration arcs have the following properties:(1) Both I(Ôf ) and I(Of ) are decomposed into l connected components as follows:I(Ôf ) =lI(α̂j  ) and I(Of ) = πf (I(Ôf )) =j  =1lI(αj  ),j  =1 
where I(α̂j  ) = ĥ−1 ({β̂j  }) and I(αj  ) = h−1 ( βj  ).
(2) The set I(Ôf ) (resp. I(Of )) is the union of all the cyclic Iˆf -components (resp. If components).
(3) For each j  modulo l , we haveΛ(α̂j  ) ∩ I(Ôf ) = Λ(α̂j  ) ∩ I(α̂j  ) = I(α̂j  ) − Ôf .
Thus we also haveΛf − I(Ôf ) =l(Λ(α̂j  ) − I(α̂j  )).

j  =1

Notation for the aﬃne laminations. For later use, we also need to translate the setsdeﬁned above to those of the aﬃne laminations in the universal setting. It is simply done by32

sending them by ιf : Anf → Af or ιg : Ang → Ag . We deﬁne as follows:
Degenerating arc system: If := ιf (Anf ∩ Iˆf ), Ig := ιg (Ang ∩ Iˆg ).
Cyclic degenerating arcs: IOf := ιf (Anf ∩ I(Ôf )).
Principal leaves: Λaf := ιf (Λf ), Λa (α̂j  ) := ιf (Λ(α̂j  )),Λag := ιg (Λg ), Λa (β̂j  ) := ιg (Λ(β̂j  )).
Note that we need to cut the irregular points out when we translate Iˆf , Iˆg and I(Ôf ).
Later we will use the fact given by Proposition 6.4(2) that IOf is the union of all cyclic
components of the degenerating arc system If .

6.3

Degeneration of the aﬃne parts/laminations.

Now we start describing the degeneration of Anf and Af . Let us check:
Proposition 6.5 (Semiconjugacy onto the aﬃne part Ang .) The preimage of the aﬃnepart Ang by ĥ : Nf → Ng is Anf − I(Ôf ). In particular, the restriction ĥ : Anf − I(Ôf ) → Ang
is a semiconjugacy. Hence we also have a semiconjugacy ĥa : Af − IOf → Ag in the aﬃnelaminations.
ˆ = {∞}
ˆ and ĥ−1 (Ôg ) = I(Ôf ). By Proposition
Proof. By Theorem 6.1 we have ĥ−1 ({∞})
n−1
ˆ and thus ĥ (Ang ) = Nf − I(Ôf ) ∪ {∞}ˆ = Anf − I(Ô).
6.2 we have Ag = Ng − Ôg ∪ {∞}Now the restriction ĥ : Anf − I(Ôf ) → Ang is a continuous and surjective map. Since the aﬃneparts and I(Ôf ) are invariant under the dynamics in the natural extensions, the restriction
is a semiconjugacy.
Under the identiﬁcation of the aﬃne parts as the aﬃne laminations of (f → g) (Proposition 6.2), we have the corresponding semiconjugacy ĥa : Af − IOf → Ag .
Meekness. As we will see, the set Anf − I(Ôf ) in the aﬃne part is an fˆ-invariant Riemannsurface leaf space. (Recall the deﬁnition of leaf spaces given in Section 3.) Thus ĥ : Anf −I(Ôf ) → Ang gives a semiconjugacy on the leaf spaces.
To describe more precise properties of this semiconjugacy, we introduce some terminologies to classify surjective and continuous maps between leaf spaces. Let L1 and L2 be leaf
spaces whose leaves are all Riemann surfaces or all hyperbolic 3-manifolds (which may be
incomplete and may have boundaries). We say• a map H : L1 → L2 is leafy if H is surjective, continuous, and for any leaf L in L2 ,there exits a unique leaf L in L1 with L = H −1 (L );• a map H : L1 → L2 is meek if H is leafy, and for any leaf L in L2 , its preimage leaf
L = H −1 (L ) is homeomorphic to L ; and• a meek map H : L1 → L2 is very meek if for any leaf L in L2 , there exists a leafwise
quasiconformal or quasi-isometric homeomorphism between L and its preimage leaf
L = H −1 (L ).
A leafy map gives one-to-one correspondence between the leaves in two leaf spaces. A meek
map H does not change the leafwise topology, but H itself need not be a leafwise homeomorphism. For example, consider the simply connected Riemann surfaces Ĉ, D, and C, which33

are all trivial leaf spaces with one leaf. Then the pinching semiconjugacy h : Ĉ → Ĉ given in
Section 5 is very meek. Any surjection H : D → C is meek, but not very meek since there isno quasiconformal map from D onto C.
Here is the main theorem in this section:
Theorem 6.6 (Aﬃne part/laminations) For any degeneration pair (f → g), the set
Anf − I(Ôf ) is a leaf space and the restriction ĥ : Anf − I(Ôf ) → Ang of ĥ = ĥn : Nf → Ng isa meek pinching semiconjugacy. More precisely,(1) Non-principal part. A further restriction ĥ| Anf − Λf → Ang − Λg is a very meek map
between the leaf spaces with leaves isomorphic to C.
(2) Principal part. The complementary restriction ĥ| Λf − I(Ôf ) → Λg is a meek map
between the leaf spaces with ﬁnitely many leaves described as follows. For each j  modulol , let L1 , . . . , Lq be the q  principal leaves in Λg (β̂j  ). Then the set Λ(α̂j  ) − I(α̂j  ) has
q  path-connected components S1 , . . . , Sq satisfying:• Each Si (1 ≤ i ≤ q  ) is a simply connected Riemann surface that is invariant underfˆl̄ .
• The preimage ĥ−1 (Li ) is Si . Thus {Si } and {Li } has one-to-one correspondence.
The corresponding semiconjugacy ĥa : Af − IOf → Ag in the aﬃne laminations also has thesame properties.
Property (1) means that any two non-principal leaves in Anf are never merged into one leaf byĥ. One can easily check that every non-cyclic degenerating arc I(ζ̂) containing ζ̂ ∈ α̂f − Ôf iscompactly contained in a leaf of Anf with respect to the leafwise topology. Then ĥ just pinchesI(ζ̂) to one point ĥ(ζ̂) ∈ Iˆg with the same type in a leaf of Ang (Figure 14). Though each nonprincipal leaf contains inﬁnitely many such Iˆf -components, the pinching map ĥ preserves the
topology of the leaves isomorphic to the complex plane. Thus the major topological change
in the degeneration is described in (2).

Figure 14: Degeneration of a non-cyclic Iˆf -component.
Figure 15 illustrates the situation of (2). As I(α̂j  ) is pinched to β̂j  , the q  componentsof Λ(α̂j  ) − I(α̂j  ) become the q  leaves of Λ(β̂j  ). Then the hyperbolic action of fˆl̄ on eachpath-connected component of Λ(α̂j  ) − I(α̂j  ) is sent to the parabolic action of ĝl̄ on each leafof Λ(β̂j  ).

34

Figure 15: A caricature of principal leaves Λ(α̂j  ) and Λ(β̂j  ) for q  = 3. Here we takesubscripts j  modulo l . Recall that q = q  and l = l in Case (a) and q = 1 < q  and l = l q  inCase (b). In Case (b), note that Λ(α̂j  ) = Λ(α̂j  +l ) = Λ(α̂j  +2l ). The heavy curves indicate
I(α̂j  ). The Rabbits are the case of l = 1.
Example: The Cauliﬂowers. (See Figure 12 again.) In this case the only cyclic Iˆf component I(Ôf ) is the invariant lift of the interval [α1 , γ1 ] by the projection πf : Nf → C̄,where γ1 is the repelling ﬁxed point. On the other hand, the corresponding set IOf in theuniversal setting is the invariant lift of (α1 , γ1 ] by the projection π̂f : Af → C̄.
The set Λf − I(Ôf ) (resp. Λaf − IOf ) is isomorphic to a plane with a half-line removed. There are many other compactly contained Iˆf -components in Λf − I(Ôf ) (resp.
If -components in Λaf − IOf ) but the map ĥ| Λf − I(Ôf ) → Λg (resp. ĥa | Λaf − IOf → Λag )just pinches them to points. Thus Λf − I(Ôf ) (resp. Λa − IOf ) has the same topology as Λg(resp. Λag ).
Example: The Rabbits. As in Figures 13 and 15, Case (a) and Case (b) exhibit diﬀerent
types of degenerations. What happens in the Case (a) Rabbits (f1 → g) is similar to the
Cauliﬂowers. The set Λf1 − I(Ôf1 ) is topologically three planes with three half-lines removed,
which are mapped onto the three principal leaves of Λg .
However, in the Case (b) Rabbits (f2 → g) the only cyclic Iˆf2 -component Ôf2 is a noncompact trivalent graph which divides the invariant leaf Λf2 into three simply connectedcomponents. These components are cyclic under fˆ2 and mapped onto the three principal
leaves of Λg .
Proof of Theorem 6.6. Let us start the proof of the theorem. We ﬁrst show:
Lemma 6.7 For any leaf L of Ang , there exists a leaf L of Anf such that ĥ−1 (L ) ⊂ L.
Moreover, ĥ−1 (L ) is path-connected.
 , . . .) and ŵ = (w , w , . . .)Proof. (cf. [LM, Lemma 3.2]) Fix two distinct points ẑ  = (z0 , z−10−1:= πg ◦ ĝ−n (η̂  ) is a path joining z−n
in L . Let η̂  be a path in L joining ẑ  and ŵ . Then η−nand w−n , and η−n with n  0 has a neighborhood U−n whose pull-back along ẑ and ŵ is
with n  0 does not pass through Og and ∞.)eventually univalent. (That is, the path η−n−1Choose any ẑ = (z0 , z−1 , . . .) ∈ ĥ (ẑ ) and ŵ = (w0 , w−1 , . . .) ∈ ĥ−1 (ŵ ). For any N ≤ 0,
passes through Ig , its preimage h−1 (η−N
) is still a path-connected set by (1) ofeven if η−N

35Theorem 5.5. Since z−N and w−N are contained in h−1 (η−N), we can choose a path η−Njoining z−N andw−N . Let us ﬁx N  0 so that η−N avoids both Og and ∞. Then η−N also
avoids I(Of ) = j  I(αj  ) and ∞. Now we can take a neighborhood of η−N whose pull-backalong ẑ and ŵ is eventually univalent. Since we can lift paths {η−N −n } to a path in Nf joiningẑ and ŵ, both ẑ and ŵ are in the same leaf L of Anf . Thus ĥ−1 (L ) is a path-connected subsetof L.
(Lemma 6.7)

It is possible that ĥ−1 (L1 ) ∪ ĥ−1 (L2 ) ⊂ L for distinct leaves L1 and L2 of Ang . In fact, it
happens when L1 and L2 are principal leaves and q = 1 < q  (Case (b)).
As we have seen, the map ĥ : Anf − I(Ôf ) → Ang is a semiconjugacy, thus surjective andcontinuous. To show that this is a leafy and meek between two leaf spaces, it is enough toshow (1) and (2).
Proof of (2).
We show (2) ﬁrst. In both Cases (a) and (b), the set Λ(α̂j  ) consistsof q principal leaves by Proposition 6.3(2) and (3). Let L be such a leaf in Λ(α̂j  ), andset L− := L − I(α̂j  ) for simplicity. The degenerating arc I(αj  ) = πf (I(α̂j  )) contains q
repelling periodic points. Let γL be the one of such repelling points so that its periodic
lift γ̂L is contained in L. Then there are q̄ := l/l (= 1 or q  according to Case (a) or(b)) degenerating arcs in I(αj  ) landing at γL downstairs, and such arcs are all invariant
under f l̄ . Since the quotient L/ fˆl̄ is isomorphic to the quotient torus of the dynamics
by f l̄ near γL = πf (γ̂L ), we conclude that the torus L/ fˆl̄ is divided into q̄ parts by q̄
simple closed curves (I(α̂j  ) − {vertices})/ fˆl̄ . This implies that L− is the union of q̄ pathconnected components divided by I(α̂j  ), say S1 , . . . , Sq̄ . (In Case (a), we have q̄ = 1 and L−
is topologically a plane minus a half line.) In particular, each Si is simply connected. Since
lq = l q  , we have q  path-connected components in Λ(α̂j  ) − I(α̂j  ) in total.
Now take L ⊂ Ang and L ⊂ Anf as in the lemma above. Suppose that ĥ(L) contains aˆ Since ĥ−1 (Ôg ) = I(Ôf ) and ĥ−1 ({∞})
ˆ = {∞},
ˆ
irregular point, that is, a point in Ôg ∪ {∞}.
it is equivalent to L ∩ I(Ôf ) = ∅. This implies that L is a principal leaf, i.e., L ⊂ Λ(α̂j  ) forsome j  modulo l . Since Ôg and L are disjoint in Ng , we haveĥ−1 (L ) ⊂ L − ĥ−1 (Ôg ) = L − I(Ôg ) = L − I(α̂j  ) = L− ,where L− is the disjoint union of S1 , . . . , Sq̄ as above. By the lemma above ĥ−1 (L ) is apath-connected subset of L− . Since each Si are also path-connected, we can ﬁnd an i withĥ−1 (L ) ⊂ Si . On the other hand, the set ĥ(Si ) is also path-connected by continuity of ĥ andthus contained in a leaf of Ang . This leaf must be L . Thus we have ĥ(Si ) = L and it implies
Si ⊂ ĥ−1 (ĥ(Si )) = ĥ−1 (L ) ⊂ Si .
Now let us check that L is one of q  leaves of Λ(β̂j  ). To identify the leaf, we consider lifts ofexternal rays. Let γ̂L be the repelling periodic point in L and set γL := πf (γ̂L ). Now there
are q̄ external rays Rf (θ1 ), . . . , Rf (θq̄ ) landing at the repelling periodic point γL downstairs,
and they all have ray-period ¯l. By comparing the quotient tori upstairs and downstairs, weconclude that each of Si has an invariant lift of such an external ray, say Rf (θ̂i ). (Where θ̂iis an cyclic angle in Θ̂ ⊂ T̂.) Since ĥ sends external rays to external rays without changing
the angles (Theorem 6.1(2)), the image Rg (θ̂i ) = ĥ(Rf (θ̂i )) is an invariant lift of a periodic
external ray downstairs which must land on the parabolic point βj  . Hence L must contain
Rg (θ̂i ) and thus L is a principal leaf in Λ(βj  ). By setting Li := L , we have the desired
correspondence between {Si } and {Li } with ĥ−1 (Li ) = Si .
36

Since the same happens for each L in Λ(α̂j  ), we have ĥ−1 (Λ(β̂j  )) = Λ(α̂j  ) − I(α̂j  ). By
taking the union with respect to j  , we have ĥ−1 (Λg ) = Λf − I(Ôf ). Since any leaf of Anf or
Ang is dense (Proposition 2.1(2)), the sets Λf − I(Ôf ) and Λg are connected, thus leaf spaces.
Now we conclude (2).
Proof of (1). Next we show (1): Suppose that ĥ(L) avoids irregular points. Then L is nota principal leaf. Since ĥ(L) is a path-connected subset of Ang , there is a leaf of Ang containingĥ(L), which must be L . It follows that ĥ(L) = L and thusL ⊂ ĥ−1 (ĥ(L)) = ĥ−1 (L ) ⊂ L.
By property (2), the leaf L is not a principal leaf. Thus the preimage of any non-principal
leaf is a non-principal leaf. Since every leaf in the aﬃne parts is dense, the non-principal parts
are also connected, thus leaf spaces. Since every leaf in the non-principal parts is isomorphic
to C, it follows that ĥ restricted on the non-principal part is very meek. This implies property(1).
Aﬃne laminations. Finally, the last part of the statement on the aﬃne laminations isjustiﬁed under the identiﬁcation given by Proposition 6.2, and by deﬁning ĥa := ιg ◦ ĥn ◦ ℘.
Remark. The aﬃne laminations Af and Ag in the universal setting have metrics induced
from a “universal metric” on Û a . Such a metric is given in [KL, §3]. By using the fact that ĥntends to identity in the natural extension, we can prove the Gromov-Hausdorﬀ convergenceof Af − IOf to Ag as f → g with respect to the universal metric.

7

Degeneration of the H3 -laminations

In this section we consider degeneration of the H3 -laminations associated with (f → g). The
main tool is a three dimensional extension of the pinching semiconjugacy ĥa : Af −IOf → Ag
in the universal setting. Unfortunately, we cannot extend ĥa to the whole vertical ﬁbersof Af − IOf without losing the semiconjugating property. However, we will see that we
can extend ĥa to a fairly large part of Hf (including the non-principal part of Hf ) still
as a semiconjugacy onto Hg , and it is useful enough to explain the degeneration of the
dynamics. More precisely, we deﬁne an fˆ-invariant set Vf (called valley) in the principal
leaves Λhf = (Λaf )h and we will have a meek pinching semiconjugacy ĥh : Hf − Vf → Hg
(Theorems 7.1 and 7.3).
In particular, we will conclude that ĥh is very meek on the non-principal part, i.e., thegeometry of the non-principal part of Hf is leafwise preserved as f → g. In the principal
part, we will see that the loxodromic (or hyperbolic) dynamics is degenerate to the parabolic
dynamics as f → g. However, the restriction of ĥh to the principal part is still meek.
The semiconjugacy above will play a crucial role when we consider the quotient laminations in the next section.
Remarks on notation. Here we will work in the universal setting. Let us recall somenotation given in Section 2.
• To denote leaves of the aﬃne laminations, we will use L or L for simplicity. (We usedthem for leaves of the aﬃne parts.) We also use φ : C → L or φ : C → L for theiruniformizations.

37• For φ : C → L above, the map πf ◦ ℘ ◦ φ : C → C̄ is an element of Kf . Recall thatπ̂f := πf ◦ ℘ on Af . Note that ψ−N := π̂f ◦ fˆa−N ◦ φ = πf ◦ fˆn−N ◦ ℘ ◦ φ determines anelement ψ̂ = (ψ−N )N ≥0 in K̂f .
• Recall that for X ⊂ Af , we denote its ﬁbers in Hf by X h . In particular, the uniformization of Lh associated with φ : C → L is denoted by φh : H3 → Lh .

7.1

Pinching semiconjugacy on H3 -laminations

Since the H3 -lamination Hf is an R+ -bundle of the aﬃne lamination Af , one may imagine
that the pinching semiconjugacy ĥa : Af − IOf → Ag is naturally extended to a surjective
map from Hf − IOhf → Hg , and this would be enough to describe the degeneration of Hf to
Hg . It is true, however, the map may not be a semiconjugacy; i.e., it may not preserve thedynamics on Hf and Hg . Such a map is useless when we consider their quotient 3-laminations.
Here we try to have the best possible 3D-extension of ĥa as a semiconjugacy. The maindiﬃculty comes from the universal setting: We need to manage plenty of complicated notation.
Pinching semiconjugacy between the non-principal parts.
extension except over the principal leaves. We ﬁrst establish:

We naturally have an

Theorem 7.1 (Non-principal part) The very meek pinching semiconjugacy ĥa : Af −
Λaf → Ag − Λag extends to a very meek semiconjugacy ĥh : Hf − Λhf → Hg − Λhg . In particular,
for x̂ ∈ Af − Λaf , the extended semiconjugacy ĥh sends the ﬁber of x̂ onto the ﬁber of ĥa (x̂).
What happens in the non-principal part is very simple: Each Ifh -component is pinched
to be an Igh -component, which is a vertical ﬁber of an Ig -component (Figure 16).

Figure 16: Degeneration of a non-periodic Ifh -component
Before the proof of this theorem, let us start with a lemma which slightly generalize [LM,Lemma 9.2]:
Lemma 7.2 (Aﬃnely natural extension) Let H : C → C be a continuous and surjective
map such that for each w ∈ C the functionvw (t) = max |H(w + W ) − H(w)||W |=tis a non-decreasing function from R≥0 onto itself. Then the map e(H) : H3 ∪ C → H3 ∪ C
deﬁned bye(H)(w, t) := (H(w), vw (t))
38

is a continuous and surjective map which restricts H on C. Moreover, the following is satisﬁed:S
1. The extension is aﬃnely natural: If δ1 and δ2 are complex aﬃne maps of C then e(δ1 )and e(δ2 ) are the unique possible similarities of H3 , ande(δ1 ◦ H ◦ δ2 ) = e(δ1 ) ◦ e(H) ◦ e(δ2 ).
2. e(H) depends continuously on H, in the compact-open topology on maps of C and H3 .
3. The vertical line over z is mapped onto to that of H(z) (but not one-to-one in general).
The proof is routine and left to the reader.
Proof of Theorem 7.1. For a leaf L of Af − Λaf , the set L := ĥa (L) is a leaf of Ag − Λag(Theorem 6.6(1)). Take uniformizations φ : C → L and φ : C → L . Then the map
H := (φ )−1 ◦ ĥa ◦ φ is a surjective and continuous map satisfying the condition of the lemma
above. (This part is also routine and left to the reader.) Let us deﬁne ĥh : Lh → Lh byĥh := φh ◦ e(H) ◦ (φh )−1 , where φh and φh are the uniformizations of H3 -leaves Lh and Lhnaturally induced by φ and φ , respectively. Then one can easily check that this deﬁnition
does not depend on the choice of φ and φ by aﬃne naturality of the extension. Moreover,the map ĥh sends a ﬁber onto a ﬁber by property 3 of the lemma above.
Now ĥh : Hf − Λhf → Hg − Λhg is deﬁned just leafwise, however, it is not hard to checkthe continuity of this map by applying property 2 of Lemma 7.2 to each local product boxof the laminations. (This part is more justiﬁed in the proof of Theorem 7.3.) Property 1 ofLemma 7.2 insures that ĥh semiconjugates fˆ to ĝ in our domain.
Finally note that Hf − Λhf and Hf − Λhf are leaf spaces by minimality ([LM, §7.7]).
Next we continuously extend ĥh above to some part of the principal leaves Λhf . We startwith carving.
Valleys carved by degenerating arcs. For an α̂j  ∈ Ôf with j  modulo l , take a leaf
L of Af contained in Λa (α̂j  ). Since any leaf is isomorphic to C, we choose a uniformization
φ : C → L, and consider a continuous function u = uφ : C → [0, ∞) deﬁned by u(w) :=distC (w, φ−1 (L ∩ IOf )), where the distance is measured in Euclidean metric on C. Set
⊂ (L − IOf )h ,Lv := φh (w, t) ∈ H3 : t < u(w)where φh : H3 → Lh is the natural uniformization induced by φ. (See Figure 17.) It is
not diﬃcult to show that the deﬁnition of Lv does not depend on the choice of φ. In fact,
for another uniformization φ1 : C → L, there exists a unique aﬃne map δ1 (w) = a + bw
such that φ = φ1 ◦ δ1 . Thus uφ (w) = uφ1 (w)/|b| and the diﬀerence is canceled when we
consider φh = φh1 ◦ e(δ1 ). We call this well-deﬁned set Lv a carved principal leaf, and thecomplement Lh − Lv the valley of Lh . (This is just like a valley carved by a degenerating“stream” IOf with a source which is the lift of a repelling point. See Figure 18.) Note thatv over all leaves in Λ(α̂  ),Lv has q̄ = l/l path-connected components. Take a union of Ljand denote it by Λ(α̂j  )v . Then the carved principal leaves Λvf := Λ(α̂j  )v and the removedpart Vf := (Λ(α̂j  )h − Λ(α̂j  )v ) are completely invariant under the action of fˆ : Hf → Hf .
Note that both Λvf and Vf have ¯l(= lq = l q  ) path-connected components. We also say Vf
is the valley of Hf . Again by minimality of leaves, Hf − Vf and Λvf are connected, thus leaf
spaces.
39

Figure 17: Carving a valley in the case of q = 1 < q  = 3, Case (b).
Examples: The Cauliﬂowers. For the Cauliﬂowers (f → g), the “stream” IOf is thehalf-line on the principal leaf Λaf . The carved principal leaf Λvf = Λhf − Vf has a fundamental
domain as shown in Figure 18, Case (a).
Examples: The Rabbits. For the Case (a) Rabbits (f1 → g) the carved principal leaves
Λvf1 are leafwise similar to the Cauliﬂower case. In particular, the set Λvf1 has three pathconnected components cyclic under fˆ1h . For the Case (b) Rabbits (f2 → g), the “stream”
IOf2 has three distinct directions but the carved principal leaf Λvf2 also has three pathconnected components cyclic under fˆ2h . See Figure 18, Case (b).
Now we are ready to state the theorem:
Theorem 7.3 (Extention to the principal part) There exists a continuous extension ofĥh : Hf − Λhf → Hg − Λhg above to a meek pinching semiconjugacy ĥh : Hf − Vf → Hg withthe following properties:(1) For each j  modulo l , there are q  path-connected components S1v , . . . , Sqv in Λ(α̂j  )v such
h
h
h

h
that Siv = ĥ−1h (Li ), where L1 , . . . , Lq  are the q principal leaves in Λ(β̂j  ) . In particular,
h
v
h
ĥ−1
h (Λg ) = Λf = Λf − Vf .

(2) For x̂ ∈ Λaf , the semiconjugacy ĥh sends {x̂}h ∩ Λvf onto the ﬁber of ĥa (x̂).
Thus we can understand the dynamics of ĝ : Hg → Hg by pinching fˆ : Hf − Vf → Hf . The
proof breaks into two main steps: First we deﬁne a semiconjugacy ĥh : Λvf → Λhg . Then weshow that it is a continuous extension of the pinching map ĥh in Theorem 7.1. To show thiscontinuity, we have to go back to the universal setting.
Proof of Theorem 7.3: Construction. We ﬁrst describe Lv of L ⊂ Λ(α̂j  ) in detail.
Let S1 , . . . , Sq̄ be the connected components of L − IOf . Each of them are sent onto leavesin Λag , say L1 , . . . , Lq̄ (Theorem 6.6(2)). Let φ and φi be uniformizations of L and Li for
i = 1, . . . , q̄. We denote φ−1 (Si ) by Si ⊂ C. Then we can deﬁne a surjective and continuous
map Hi : Si → C by Hi := (φi )−1 ◦ ĥa ◦ φ. For each w ∈ Si , set u(w) = distC (w, ∂Si ) and wedeﬁne vw : [0, u(w)) → [0, ∞) byvw (t) = max |Hi (w + W ) − Hi (w)|,|W |=t40

Figure 18: A fundamental domain of the action fˆl̄ : Λvf → Λvf is mapped onto that ofĝl̄ : Λhg → Λhg by the semiconjugacy ĥ.
which is a non-decreasing function from [0, u(w)) onto [0, ∞). Set

Svi := (w, t) ∈ H3 : w ∈ Si , t < u(w) andSiv := φh (Svi ).
Note that Lv is the disjoint union of Siv for i = 1, . . . , q̄. Now it is not diﬃcult to checkthat there exists a continuous extension of Hi : Si → C to e(Hi ) : Svi → H3 . Moreover, thisextension is aﬃnely natural: That is, property 1 of Lemma 7.2 holds over δ2−1 (Si ). We deﬁnehh −1ĥh : Siv → Lhi by ĥh := φi ◦ e(Hi ) ◦ (φ ) . By aﬃne naturality, one can check that this map
does not depend on the choice of the uniformizations φ and φi ; and that ĥh semiconjugates
 
 
h
v
h
the actions of fˆl q : Siv → Siv onto ĝl q : Lhi → Li . Now we have ĥh : Λf → Λg that isdeﬁned on each path-connected components.
Continuity. Next we prove that the map ĥh : Hf − Vf → Hg is continuous; that is, for any
sequence x̃n → x̃ within Hf − Vf , the sequence ỹn := ĥh (x̃n ) converges to ỹ := ĥh (x̃).
To characterize the convergence in Hf − Vf , we need to start
Convergence in Hf .
with a lot of notation. For each x̃n ∈ Hf − Vf , take a representative (ψn,−N )N ≥0 in K̂f .
Set x̂n := pr(x̃n ) ∈ Af and ẑn = (zn,−N )N ≥0 := ℘(x̂n ) ∈ Anf . Let Ln denote the leafιf (L(ẑn )) in Af containing x̂n . Then there exists a uniformization φn : C → Ln such that
ψn,−N = π̂f ◦ fˆa−N ◦ φn . For x̂n ∈ Ln , let Sn be the connected component of Ln − IOf
containing x̂n . It may be a sector when x̃n ∈ Λvf , or whole a leaf Ln when x̃n ∈/ Λvf . Set
Sn := φ−1n (Sn ). We also deﬁne the function uφn : C → [0, +∞] by uφn (w) := distC (w, ∂Sn ) ifx̃n ∈ Λvf , and by uφn (w) ≡ +∞ otherwise.
For x̃, we also deﬁne the following sets just by removing n’s from the deﬁnitions above: arepresentative (ψ−N )N ≥0 in K̂f , the projection x̂ in Af , a leaf L containing x̂, the uniformization φ : C → L induced by (ψ−N )N ≥0 , the path-connected component S ⊂ L containing x̂,S = φ−1 (S), and the function uφ : C → [0, +∞].
Since x̃n → x̃, we may assume that ψn,−N → ψ−N uniformly on compact sets for any
ﬁxed N . Note that for the uniformizations φhn : H3 → Lhn and φh : H3 → Lh , both x̃n and41

x̃ have coordinate (0, 1) with respect to ψn and ψ: i.e., (φhn )−1 (x̃n ) = (φh )−1 (x̃) = (0, 1).
Moreover, both uφn (0) and uφ (0) are greater than 1 since x̃n , x̃ ∈ Hf − Vf .
Here is an important lemma for the continuity:
Lemma 7.4 Set Cn := uφn (0) and C := uφ (0). Thenlim inf Cn ≥ C.
n→∞

Proof of Lemma 7.4. First we consider the case of x̃ ∈ Λvf , thus C is ﬁnite. Since/ Λvf , it is enough to show the case of x̃n ∈ Λvf for all n  0.
Cn = +∞ when x̃n ∈Take a disk D̃ := DC̃ with := C/C̃  1. If N  0, the map ψ−N : D̃ → C̄ is univalent.
Moreover, the sequence ψn,−N |D̃ converges uniformly to ψ−N |D̃ and thus is univalent for alln  0. Set B := ψ−N (D2C ) ⊂ C. (Since f is a polynomial, inﬁnity is the exceptional value
for all elements of Kf .) Then B ∩ I(Of ) = ∅ since D2C ∩ ∂Sn = ∅. In particular, there isa point in B ∩ I(Of ) that attains distC (z−N , B ∩ I(Of )). Since zn,−N = ψn,−N (0) tends toz−N = ψ−N (0) ∈ B, we have
distC (zn,−N , B ∩ I(Of )) → distC (z−N , B ∩ I(Of )).
By the Koebe distortion theorem,
Cn = |(ψn,−N ) (0)|−1 distC (zn,−N , B ∩ I(Of )) (1 + O( ))C = |(ψ−N ) (0)|−1 distC (z−N , B ∩ I(Of )) (1 + O( )).
This implies that Cn is arbitrarily close to C as n → ∞ if we take C̃, N  0 and  1 asabove. Hence we have lim Cn = C.
Next we consider the case of x̃ ∈/ Λvf , that is, C = +∞. Suppose that lim inf Cn = M <+∞. Passing through a subsequence, we may assume that Cn → M and x̃n ∈ Λvf for alln  0. By a similar argument as above, we have
distC (z−N , ψ−N (D2C ) ∩ I(Of )) ∼ distC (zn,−N , B ∩ I(Of ))∼ M |(ψn,−N ) (0)|∼ M |(ψ−N ) (0)|for all N, n  0. Since (ψ−N −m ) (0) = (ψ−N ) (0)/(f m ) (z−N −m ) for all m > 0, we havedist(z−N −m , ψ−N −m (D2C ) ∩ I(Of ))  |(ψ−N −m ) (0)| → 0 (m → ∞)by hyperbolicity of f . This implies ẑ = (z−N )N ≥0 accumulates on I(Of ) but it is possible/ Λvf .
(Lemma 7.4)
only if x̂ ∈ Λaf . It contradicts x̃ ∈Convergence in Hg . For ỹn and ỹ in Hg , we also deﬁne the following as we did for x̃:)N ≥0 and (ψ−N)N ≥0 in K̂g , the projections ŷn and ŷ in Ag , leaves Lnrepresentatives (ψn,−N
and L of Ag containing ŷn and ŷ, and their uniformizations φn : C → Ln and φ : C → L
induced by the representatives. One can easily check the following properties:(1) ŷn = ĥa (x̂n ) and ŷ = ĥa (x̂). Thus ŷn → ŷ in Ag by continuity of ĥa .
(2) Ln = ĥa (Sn ) and L = ĥa (S).
= π̂g ◦ ĝa−N ◦ φn and ψ−N= π̂g ◦ ĝa−N ◦ φ .
(3) ψn,−N42Moreover, property (1) insures that we can choose the representatives satisfying ψn,−N→
ψ−N for any ﬁxed N as n → ∞. Suppose that ỹn and ỹ have coordinates (0, ρn ) and (0, ρ)
with respect to ψn and ψ  respectively. Now it is enough to show that ρn → ρ for thecontinuity of ĥh .
Let Hn : Sn → C and H : S → C be surjective maps deﬁned by Hn := (φn )−1 ◦ ĥa ◦ φnand H := (φ )−1 ◦ ĥa ◦ φ. Now we have the following diagram:
ˆ−N

π̂f
φn
fa
−→ fˆa−N (Sn ) −−−−→Sn −−−−→ Sn −−−






Hn 
ĥa
ĥa

C

φ

ĝa−N

C̄


h

π̂g

−−−n−→ Ln −−−−→ ĝa−N (Ln ) −−−−→ C̄.

By property (3) and this diagram, we have
◦ Hn = h ◦ ψn,−N and ψ−N◦ H = h ◦ ψ−N .
ψn,−NTake an  1 and an N  0 such that 1 + < C = uφ (0) and ψ−N : D1+ → C is univalent.
By Lemma 7.4, the disk D1+ is contained in Sn for all n  0. Moreover, the sequence ψn,−N
converges to ψ−N uniformly on D1+ and they are all univalent on it. By the dynamics of fand ψ−Nare also univalent on Hn (D1+ ) and H(D1+ )and g related by h, this implies ψn,−N
respectively. Thus we have)−1 ◦ h ◦ ψn,−N
Hn = (ψn,−N−→ H = (ψ−N)−1 ◦ h ◦ ψ−N

uniformly on D1+ . By deﬁnition of ĥh , we haveρn = max |Hn (w + W ) − Hn (w)| and ρ = max |H(w + W ) − H(w)|,|W |=1
w=0|W |=1
w=0

and thus ρn → ρ. This completes the proof of Theorem 7.3.

Remark. Again Hf and Hg have metrics induced from a universal metric on Û h . The
Gromov-Hausdorﬀ convergence of Hf − Vf to Hg can be proved by using a similar idea as
the proof above and the fact that h tends to identity.

8

Degeneration of the quotient 3-laminations

From this section we will mainly deal with the quotient laminations and their ends. (Recall
the settings of Section 4.)
Abstract of this section. For our degeneration pair (f → g), the meek pinching semiconjugacy ĥh : Hf − Vf → Hg in the previous section provides us with a natural meek pinching
map ĥq : Mf − v f → Mg , where v f = Vf /fˆ is the quotient valley (Theorem 8.2). Since ĥq
is very meek in the non-principal part, we conclude that the geometric diﬀerence of Mf andMg appears only in the principal parts. Indeed, the unique principal leaf of Mf is a solid
torus, but that of Mg is a rank one cusp. However, they are both homeomorphic to D × T,
thus we will also conclude that the leafwise topology of Mf is preserved as f → g.
In addition, we check that the map ĥq : Mf − v f → Mg extends to their conformalboundaries.

43

8.1

Pinching map on the quotient 3-laminations

For our degeneration pair (f → g), we consider the quotient 3-laminations Mf = Hf /fˆhand Mg = Hg /ĝh , and give a description of Mg by translating the topological/geometricproperties of Mf .
For any x̃ ∈ Hf and any integer m ∈ Z, the quotient by fˆh identiﬁes fˆhm(x̃) and x̃. We
denote this identiﬁed class by x = [x̃]. We will generally use x or y for points of Mf , and 
for a leaf of Mf .
Solid tori and rank one cusps. Let us brieﬂy recall two kinds of hyperbolic 3-manifolds
with π1 isomorphic to Z. We regard the hyperbolic 3-space H3 as the set of all (z, t) ∈ C × R+
with hyperbolic metric ds2 = (|dz|2 + dt2 )/t2 . A solid torus (or a handle body) is a hyperbolic
3-manifold isomorphic to H3 / (z, t) → (λz, |λ|t) , where |λ| > 1. The core curve of a solid
torus is the only closed geodesic corresponding to the t-axis. A rank one cusp is a hyperbolic
3-manifold isomorphic to H3 / (z, t) → (z + 1, t) . A rank one cusp has non-trivial closed
curves with arbitrarily short length. Note that both solid tori and rank one cusps have thesame topology homeomorphic to D × T.
Principal leaves and valley. The quotient laminations can be divided into the principal
parts and the non-principal parts. Let us characterize the leaves in these parts:
Proposition 8.1 (Quotient leaves) For any degeneration pair (f → g), the leaves in thequotient laminations are classiﬁed as follows:• Non-principal part. For any non-principal leaf Lh of Hf (thus in Hf − Λhf ), itsquotient leaf is isomorphic to either a solid torus or H3 according to Lh is periodic or
non-periodic. The same is true for the non-principal leaves of Hg .
• Principal part. The quotient f := Λhf /fˆh is a solid torus and g := Λhg /ĝh is a rank
one cusp.
Note that even if f (or g) has two or more principal leaves, its quotient is only one leaf.
Proof. The statement about the non-principal parts is straightforward since periodic points
in the non-principal parts are lifts of repelling periodic points downstairs.
Any principal leaf Lh in Λhf is invariant under the iteration of fˆl̄ = fˆhl̄ where ¯l = lq = l q  .
Since fˆl̄ : Lh → Lh is a hyperbolic action (Proposition 6.3(5)) and every point in Λh cyclicallyflands on Lh , the quotient f of Λhf is isomorphic to a solid torus with leafwise ideal boundary
isomorphic to the torus L/fˆl̄ . (Note that the conformal boundary of f is diﬀerent. See thearemark below.)
Similarly, any leaf Lh in Λhg has period ¯l too and the action ĝl̄ : Lh → Lh is conjugate
to a translation in H3 . Thus the quotient g of Λhg is isomorphic to a rank one cusp whose
leafwise ideal boundary is isomorphic to L /ĝal̄ , or the cylinder C/Z.
We also say f and g the principal leaves of Mf and Mg respectively. Since any component of Vf ⊂ Λhf is invariant under fˆl̄ , its quotient v f := Vf /fˆ is well-deﬁned. We say v f
is the valley of f .
Remark on the boundaries of leaves. For any leaf  in Mf , its leafwise ideal boundaryand global conformal boundary are diﬀerent objects. For example, consider the repelling ﬁxed
point z = 1 of f0 (z) = z 2 and the invariant leaf L0 of Af0 corresponding to the backward
orbits converging to z = 1. Then the action fˆ0 : L0 → L0 is conformally conjugate to44

w → 2w. By taking the quotient of its 3D extension fˆ0 = fˆ0h : Lh0 → Lh0 , we have a leaf
0 of Mf0 isomorphic to a solid torus whose ideal boundary is conformally isomorphic to
the torus C∗ / w → 2w . But this ideal boundary contains points in the Julia set Jf0 . The
global conformal boundary ∂0 which is actually visible from Mf0 is two annuli isomorphic to
H/ w → 2w and H− / w → 2w . According to the product structure described as in Section
4, it is natural to draw 0 := ∂0 ∪ 0 as (an annulus)×[0, 1] like Figure 19.

Figure 19: The leafwise ideal boundary of 0 (at the lower left) and its conformal boundaryin Mf0 (at the lower right). The diﬀerence comes from how we represent a solid torus: As
D × T, or as (an annulus)×(0, 1). The red curves indicate the lifted Julia set.
Pinching map on the quotient 3-laminations.
theorem in this section:

Now we are ready to state the main

Theorem 8.2 (Quotient 3-laminations) There exists a meek map ĥq : Mf − v f → Mg
such that(1) Non-principal part: The restriction ĥq | Mf −f → Mg −g is very meek. In particular,
for any non-principal leaf  of Mf ,  and ĥq () are both isomorphic to H3 , or both
isomorphic to solid tori.
(2) Principal part: The complementary restriction ĥq | f − v f → g is meek. In particular,
f , f − vf and g are all homeomorphic.
Thus the leafwise topology of Mf is preserved as f → g. (Note that this theorem is only
about the interior of Mf ; not about Mf .) By (1), the global laminar structure is preserved
at least in the non-principal part.
As to be expected by the observation of Section 4, we would ﬁnd the major topologicaldiﬀerence of Mf and Mg in the conformal boundaries, in particular, in the lower ends. Thisis the main topic of the next two sections.
Proof of Theorem 8.2: Existence.
We construct such a map ĥq from the pinching
semiconjugacy ĥh : Hf − Vf → Mg . Take x̃ and ỹ in Hf − Vf with ỹ = fˆn (x̃) for some n.
Then ĥh (ỹ) = ĥh ◦ fˆn (x̃) = gn ◦ ĥh (x̃) and thus [ĥh (x̃)] = [ĥh (ỹ)] in the quotient Mg . Thisimplies that a map ĥq : Mf − v f → Mg is induced from ĥh : Hf − Vf → Hg by deﬁningĥq ([x̃]) := [ĥh (x̃)].

45

Note that both Mf − v f and Mg are leaf spaces since Hf − Vf and Hg are. Thus themap ĥq : Mf − v f → Mg is a surjective and continuous map between leaf spaces. To showthat ĥq is meek, we divide the map into the non-principal part ĥq |Mf −f and the principal
part ĥq |f −vf , and check their meekness separately. (Note also that Mf − f is a leaf spaceand  − v f is a leaf space with one path-connected component.)(1): Non-principal part. Now we can describe the topology of Mg by pinching Mf .

Take any non-principal leaf  of Mg . By Theorem 7.1, its preimage  := ĥ−1q ( ) must be also
a non-principal leaf, and by deﬁnition, the map ĥq :  →  just pinches Ifh /fˆ-components of to Igh /ĝ-components of  . Since Ifh has no periodic component in the non-principal leaves,each Ifh /fˆ-component in  is topologically a star-like graph times R+ where R+ corresponds tothe direction of the vertical extension from Af to Hf . (cf. Figure 16.) When  is isomorphic
to H3 , so is  since ĥq :  →  sends the vertical ﬁbers to the vertical ﬁbers. When  is a solid
torus, it is the quotient of a non-principal periodic leaf Lh in Hf . Since ĥh is a semiconjugacy,
 is also the quotient of non-principal periodic leaf ĥh (Lh ), thus a solid tori.

Figure 20: A non-principal solid torus in Mf is pinched to a solid torus in Mg . The blue
curves show the core curves in the solid tori.
Remark. This argument would be more justiﬁed by considering the leafwise homotopy:When  is a non-principal leaf in Mg , the leaf  = ĥ−1q ( ) is homotopy equivalent to  . Thus3
 and  must be the same type  H or solid tori.
When  is a solid torus, one can see the vertical ﬁbers of Lh in  as a one-dimensional
foliation with ﬁbers coming from the ideal boundary and coiling around the core curve of .
(See Figure 20.) Even in this case, the pinching map ĥq :  →  does not change the topology
and  is still a solid torus.
(2): Principal part.
By Theorem 7.3(1), we have ĥ−1q (g ) = f − v f . As Figure 21
indicates, the valley vf and f are topological solid tori sharing a circle at their boundary.
In particular, f − vf is topologically a solid torus D × T too. (Note that the ideal/conformal
46

boundaries are not included in this argument.) The principal leaf g is a rank one cusp, but
it is also homeomorphic to D × T.
Figure 21: A topological picture of the degeneration of the principal leaf f to g by ĥq . The
valleys are shown in green, and the blue dotted curves show the core curves.
Remarks on Figure 21: Generation of the cuspidal part. On the left of Figure 21,
a leafwise view with ideal boundaries, the valley v f is shown as a solid torus. On the right,
quasi-Fuchsian-like view with conformal boundaries, v f is shown as (an annulus)×(an open
interval). By the pinching map ĥq , the valley v f in f is pushed away to the cuspidal part ofthe lower end.
The Ifh /fˆ-components in f should be also drawn as in Figure 20. (We abbreviate themfor simplicity.) On the other hand, the Igh /ĝ-components in g are a little diﬀerent fromthose in Figure 20. Since g do not have the core curve, any Igh /ĝ-component in g escapes tothe cuspidal part of the lower end. We may consider that the coiling Ifh /fˆ-components arepushed away with the valley to the cuspidal part of g by ĥq .
Note also that the lower end may be much more complicated in general. (cf. The Rabbits,
see Figure 29.)

8.2

Extending the pinching map to the Kleinian laminations

Next we extend the meek pinching map ĥq : Mf − vf → Mg to their conformal boundaries.
We need some more notation:
Degenerating arcs in the conformal boundary. To describe the degeneration of theconformal boundary ∂Mf = Ff /fˆ, the set If of degenerating arcs again plays an essential
role.
47

Since we have to take a quotient in the Fatou set Ff , we need to cut out the part of If
in the Julia set. We set
Zf := If ∩ Ff .
Since Bf does not intersect with If , the set Zf is a subset of Df . (Recall that Ff = Bf  Df ,deﬁned in Section 4.) The action fˆa on Zf is homeomorphic and properly discontinuous.
Thus the quotientz f := Zf /fˆalives in the lower end Sf = Df /fˆa . We also call the path-connected components of Zf or z f(“Zf -components” or “z f -components” for short) the degenerating arcs in Df or Sf .
Since the restriction ĥa | Ff −Zf → Fg of ĥa : Af −IOf → Ag is a conjugacy by Theorems
6.1 and 6.6, we have a homeomorphism ĥq : ∂Mf − z f → ∂Mg . Hence:
Theorem 8.3 (Pinching map on the Kleinian laminations) The meek pinching map
ĥq : Mf − v f → Mg extends to a meek map ĥq : Mf − v f ∪ z f → Mg between two leaf
spaces whose leaves are 3-manifolds with boundaries.

9

Degeneration of the lower ends

In the following two sections, we observe topological changes of the Kleinian lamination as
the parameter c of fc moves from one hyperbolic component to another, via parabolic g = fσ .
This motion corresponds to a pair of degeneration and bifurcation processes of hyperbolic
quadratic maps like the Rabbits.
The main tool is again tessellations associated with a degeneration pair (f → g).
Abstract of this section. This section is devoted for the degeneration process. Our aim is
to complete a topological and combinatorial description of the Kleinian laminations associatedwith parabolic quadratic maps. By Theorem 8.2, it is enough to study the structures of theconformal boundaries. In particular, we may focus on the lower ends by Proposition 4.1.
First we stand on the fact that any parabolic quadratic map g = fσ has a Case (a)degeneration pair (f → g). To describe the structures of the lower ends Sf = Df /fˆa and
Sg = Dg /ĝa for this (f → g), we will realize fundamental regions QP f and QP g of the actions
fˆa : Df → Df and ĝa : Dg → Dg by means of panels in the tessellations. Then we will obtain
detailed topological descriptions of the lower ends. For example, we show that Sf is compact
but Sg is non-compact (Proposition 9.6). As a corollary, the Kleinian lamination Mf cannot
be homeomorphic to Mg .
In addition, we deﬁne the quotient tessellations Tessq (f ) and Tessq (g) supported on Sf
and Sg respectively. Then we will check that the combinatorics of the tiles in the quotient
tessellations are the same (Proposition 9.8).
Finally, as an example we apply these results to the Cauliﬂowers.
Note. The structure of Sf can be described in a diﬀerent way by applying the argument of[LM, §§3.5], but we would need an extra care for Sg , the parabolic case. In this section wegive a method that can deal with both Sf and Sg .

9.1

Kleinian laminations for parabolic quadratic maps

So far we have seen that the topologies of the laminations of general parabolic g mostly
inherit those of f , where f is a perturbation of g such that (f → g) is a degeneration pair. In
particular, the leafwise topology of the quotient 3-lamination Mf is preserved as f → g by48

Theorem 8.2. According to the analogy of the quasi-Fuchsian example (Section 4), we turnto the study of the Kleinian laminations Mf and Mg to detect the diﬀerence of Mf and
Mg . Here we make a thorough study of the combinatorial and topological properties of thelower ends for such a degeneration pair.
Assumption (a). To simplify the arguments, we make a technical assumption. Here is adirect corollary of [Mi, Lemma 4.4] (cf. [Ka2, Theorem 1.1]):
Proposition 9.1 For any parabolic quadratic map g, there exists a Case (a) degeneration
pair (f → g), that is, q = q  and l = l .
In the rest of this section we stand on this fact and assume that (f → g) is Case (a). (Wewill refer this assumption as “Assumption (a)”.) Note that in the tessellation Tess(f ) the
degenerating edge of Tf (θ, m, +) is shared by Tf (θ, m, −) under this assumption. (See [Ka3,
Proposition 3.2] and Figure 10.)
The following argument also works for Case (b) (the case of q = 1 < q  ) with a slight
modiﬁcation. But we will deal with Case (b) in the next section as a bifurcation from g.

9.2

Fundamental region of the lower ends

Tessellations over Df and Dg . To describe the topology of Sf and Sg in detail, weconstruct fundamental regions of them in Df and Dg by means of tessellation.
Let us check some basic facts on the tessellations Tessa (f ) and Tessa (g) on the aﬃnelaminations. A tile in Tessa (f ) is deﬁned by
Tfa (θ̂, m, ∗) := ιf Tf (θ̂, m, ∗)where Tf (θ̂, m, ∗) is a tile in Tessn (f ) and ιf : Anf → Af sends the objects in the naturalextension to the universal setting (Section 6). By Πaf (θ̂, ∗) we denote the union of all possible
tiles of the form Tfa (θ̂, m, ∗) with m ∈ Z. Now the set Df − Zf is tessellated by the tiles andthe panels of Tessa (f ), sinceπ̂f−1 (Kf◦ − If ) = Df − If = Df − Zf .
We apply similar notations for the lifted tessellation of Dg = π̂g−1 (Kg◦ ).
Recall that the set Θ̂ = Θ̂(f →g) is an invariant subset of the lifted angle doubling map
δ̂ : T̂ → T̂. For any θ̂ ∈ Θ̂, tiles and panels satisfyfˆa (Tfa (θ̂, m, ∗)) = Tfa (δ̂(θ̂), m + 1, ∗)and fˆa (Πaf (θ̂, ∗)) = Πaf (δ̂(θ̂), ∗).

There are only ﬁnitely many cyclic panels in the tessellation, which are associated with cyclic
angles in Θ̂.
Cyclic panels vs. non-cyclic panels. The fundamental regions we will construct are
based on the dynamics of the panels. We will divide the fundamental region into two parts:
The cyclic part Qf (±) and the non-cyclic part Pf (±), which correspond to the cyclic panels
and the non-cyclic panels respectively. To construct them, let us start with some deﬁnitions
on special angles.
A cyclic angle θ0+ (downstairs). We ﬁrst ﬁx a cyclic angle θ0+ in Θ so that we havea basepoint of the eventually periodic dynamics δ : Θ → Θ. We choose θ0+ as follows: Let49

β0 be the parabolic periodic point in Og that is on the boundary of the Fatou component
containing the critical point 0. In the type Θ(β0 ), there exist two angles θ0+ and θ0− with¯
representatives θ0+ < θ0− ≤ θ0+ + 1 such that δl (θ0± ) = θ0± and ∂Πg (θ0± , ±) contains 0. (Suchangles are characterized by the critical sector in [Ka3, §2.3]).
Examples. For the Cauliﬂowers, we set θ0+ = 0 and θ0− = 1 ≡ 0. For the Rabbits, we set
θ0+ = 4/7 and θ0− = 8/7 ≡ 1/7. For the Airplains, we set θ0+ = 5/7 and θ0− = 9/7 ≡ 2/7.
Now we have:
Proposition 9.2 Every panel in Tess(f ) or Tess(g) is uniquely parameterized by an angle
that eventually lands on θ0+ and signature ∗ = ±. In addition, under Assumption (a), thecyclic panel of angle θ0+ and signature ∗ consists of the tiles with addresses of the form
(θ0+ , µl, ∗) (µ ∈ Z).
Proof. Since the combinatorics of tessellations are the same for f and g (Section 5), it isenough to show the statement for g.
If q = q  = 1 and l > 1 (like the Airplains), the angles θ0+ and θ0− belong to the diﬀerent
cycles under angle doubling but Πg (θ0+ , ∗) = Πg (θ0− , ∗) by deﬁnition. Indeed, each panel has
two angles in Θ = Θ(f →g) that eventually land on either θ0+ or θ0− . (See “Remark on angles
and levels” in [Ka3, §§3.2].) Otherwise θ0+ and θ0− belong to the same cycle. Each panel has aunique angle in Θ that eventually lands on θ0+ . Hence in both cases, every panel is uniquelyparameterized by an angle that eventually lands on θ0+ and signatures ∗ = ±.
By deﬁnition of tiles in [Ka3, §3] and Assumption (a), the panel Πg (θ0+ , ∗) consists of thetiles with addresses of the form (θ0+ , µl, ∗) (µ ∈ Z).
Fundamental angles (upstairs).
Next we consider the “fundamental angles” of the+)n≥0 ∈ Θ̂ denote the cyclic lift of θ0+ ; that is,
dynamics δ̂ : Θ̂ → Θ̂ upstairs. Let θ̂ + = (θ−n+= θ0+ for all k ≥ 0. Then every cyclic panel in Tessa (g) lands on either Πag (θ̂ + , +) orθ−kl̄Πag (θ̂ + , −) .
We say ω̂ = (ω−n )n≥0 is an element of Ω ⊂ T̂ if ω0 = θ0+ and ω−n = θ0+ for any n ≥ 1.
Note that Ω is a compact subset of Θ̂. Let θ̂ = (θ−n )n≥0 ∈ Θ̂ be a non-cyclic angle such thatthe forward iteration {δm (θ0 )}m>0 eventually lands on θ0+ . Then there exist unique n ∈ Z
and ω̂ ∈ Ω such that δ̂n (θ̂) = ω̂ and ĝn (Πag (θ̂, ∗)) = Πag (ω̂, ∗). Since the panels in Tessa (f )also have the same properties, we have:
Proposition 9.3 The quotient of each non-cyclic panel is uniquely represented by a panel
with an angle in Ω. If θ̂ = (θ−n )n≥0 is θ̂ + or it belongs to Ω as above, then θ0 = θ0+ . Thus
the panel of angle θ̂ and signature ∗ upstairs is projected onto the panel of angle θ0+ and thesignature ∗ downstairs. Moreover, this panel upstairs consists of tiles with address of theform (θ̂, µl, ∗) (µ ∈ Z).
Fundamental regions: Cyclic parts. Let us deﬁne fundamental regions of the cyclic
panels in Df and Dg . One can ﬁnd such regions in Πaf (θ̂ + , ∗) and Πag (θ̂ + , ∗) as follows: Set
q−1

q−1

Tfa (θ̂ + , µl, ∗)Qf (∗) :=Tga (θ̂ + , µl, ∗)and Qg (∗) :=

µ=0

µ=0

for ∗ = + or −. Here the closures of tiles in Qf (∗) are just tiles plus their degenerating edges
on IOf . (That is, we take the leafwise closures.) By Assumption (a), Qf (+) and Qf (−)
50

share their degenerating edges as in Figure 22 (upper left). Note that for f (resp. g) the
principal leaf containing the lifted external ray of angle θ̂ + also contains Qf (∗) (resp. Qg (∗)).
By properties of the panels of angle θ̂ + , one can easily check that every orbit in the cyclic
panels passes through those set just once, except some orbits landing on the boundaries.

Figure 22: A caricature of Qf (∗) and Qg (∗) (in green) for q  = 3 under Assumption (a). The
dashed arrow show the actions of f l̄ and gl̄ downstairs, and those of fˆl̄ and ĝl̄ upstairs. The
stars show the critical points z = 0 downstairs, and their lifted images upstairs. The blue
doted lines are the external rays.
Fundamental regions: Non-cyclic parts. Next one can choose fundamental regions of
the non-cyclic parts in Df and Dg as follows: For each ∗ = ±, setΠaf (ω̂, ∗) − {γfa (ω̂)} and Pg (∗) :=Πag (ω̂, ∗),Pf (∗) :=
ω̂∈Ω

ω̂∈Ω

where γfa (ω̂) := ιf (γf (ω̂)) ∈ Jf . By Assumption (a), panels in Pf (+) share the degeneratingedges with those in Pf (−) (Figure 23, upper left). Note that each Πaf (ω̂, ∗) contains one

point of π̂f−1 (α0 ), since π̂f : Af → Ĉ projects Πaf (ω̂, ∗) over the panel Πf (θ0+ , ∗) downstairs,
and this panel contains α0 on its boundary. Note also that for each ∗ = ±, topologically theinterior of Pf (∗) is the product of the disk Πf (θ0+ , ∗)◦ downstairs and the Cantor set Ω. (Thesame holds for the interior of Pg (∗).)
By properties of the panels with angles in Ω, every orbit in the non-cyclic panels passes
through those set just once except the orbits landing on the boundaries of panels. Indeed,such exceptional orbits land on these boundaries at most ﬁnite times.
By these construction of Qf (∗) and Pf (∗), we conclude:
Proposition 9.4 (Fundamental regions) The sets
( Qf (∗) ∪ Pf (∗) ) and QPg :=QPf :=∗=±( Qg (∗) ∪ Pg (∗) )∗=±51

Figure 23: A caricature of Pf (∗) and Pg (∗) for q  = 3. Note that they are projected ontoΠf (θ0+ , ∗) − γf (θ0+ ) and Πg (θ0+ , ∗) downstairs. The stars show the critical point z = 0. The
doted lines are the external rays of angles θ0± .
are fundamental regions of Sf = Df /fˆa and Sg = Dg /ĝa in the dynamics on Df and Dg . In
particular, ĥa sends QPf − Zf homeomorphically onto QPg .

9.3

Topology of the lower ends

Let us describe the topology of Sf = Df /fˆ and Sg = Dg /ĝ by means of the fundamental
regions QPf and QPg . For ∗ = ±, we set Q∗f := Qf (∗)/fˆ, Q∗g := Qg (∗)/ĝ, Pf∗ := Pf (∗)/fˆ
and Pg∗ := Pg (∗)/ĝ for simplicity. Then it is clear by construction that

Sf = QPf /fˆ =Q∗f ∪ Pf∗and Sg = QPg /ĝ =∗=±
Q∗g ∪ Pg∗ .

∗=±Relations between P and Q.
The lower ends Sf and Sg have both diﬀerences andcommon properties. Let us ﬁrst check some common properties to Sf and Sg :Proposition 9.5 (P winds around Q) In the lower ends Sf and Sg , the subsets Q∗f , Pf∗
Q∗g , and Pg∗ (∗ = ±) satisfy the following:(1) For each ∗ = ±,Q∗f ∪ Pf∗ = Pf∗andQ∗g ∪ Pg∗ = Pg∗ .

(2) Pf+ ∪ Pf− = Sf and Pg+ ∪ Pg− = Sg .
Proof. Fix a signature ∗ = ± and take any point in x̂ ∈ Qf (∗) ⊂ Df . We ﬁrst show thata∗∗for every ω̂ ∈ Ω there exists a sequence x̂n ∈ Πf (ω̂, ∗) with [x̂n ] → [x̂] in Sf , thus Qf ⊂ Pf .
a
Let us divide Πf (ω̂, ∗) into a countable union n∈Z Q−n given by
Q−n :=

Tfa (ω̂, µl − n¯l, ∗) : 0 ≤ µ ≤ q − 152so that π̂f (Qf (∗)) = π̂f (fˆnl̄ (Q−n )). Since δ̂nl̄ (ω̂) → θ̂ + as n → ∞, we can approximate any
point x̂ ∈ Qf (∗) by points x̂n in fˆnl̄ (Q−n ). Set x̂n := fˆ−nl̄ (x̂n ). Then we have the sequencex̂n ∈ Q−n with [x̂n ] → [x̂] as desired. Note that [Q−n ] (tiles with deeper levels) are folding
to Q∗f as n → ∞.
Next we take any point x in Pf∗ − Pf∗ . Then there exists a sequence x̂n ∈ Pf (∗) which
has no limit point in Pf (∗) but [x̂n ] → x in the quotient. By construction of Pf (∗), such asequence must accumulate on {γfa (ω̂) : ω̂ ∈ Ω} ⊂ Jf . (Recall that Ω is compact.) It implies
that x̂n are contained in tiles with deeper levels as n → ∞. Thus [x̂n ] must accumulate onQ∗f and it follows that x ∈ Q∗f . This completes the proof of (1) for f .
Since Qf (∗) ∪ Pf (∗) is a fundamental region for Sf , (2) immediately follows from (1).
These arguments clearly works if we replace f by g.

By this proof it follows that any quotient panel [Πaf (ω̂, ∗)] in Pf∗ accumulates on Q∗f in Sf .
On the left of Figure 24 a one-dimension down caricature of this situation is shown. The
central thick circle is Q∗f with laminated quotient panels in Pf∗ winding around. The same istrue for g.

Figure 24: Laminar structures of Sf =Compactness of the lower ends.
following:∗∗∗=± (Pf ∪ Qf ) (or Sg =∗∗∗=± (Pg ∪ Qg )).

A signiﬁcant diﬀerence between Sf and Sg is the

Proposition 9.6 (Compactness) The lower end Sf is compact but Sg is non-compact.
Proof. Take any sequence x̂n in Pf (∗) escaping from all compact subsets of Pf (∗). By
construction of Pf (∗), the sequence [x̂n ] must accumulate on Q∗f by Proposition 9.5. SinceQ∗f is compact, we have the compactness of Sf . On the other hand, similar sequence x̂n in
Pg (∗) may get closer and closer to Q∗g in the quotient, but it may not accumulate since Q∗g is53

non-compact. In fact, we have such an escaping sequence [x̂n ] ∈ Sg by taking x̂n ∈ Pg (∗) sothat the heights of zn = π̂g (x̂n ) are unbounded with respect to a ﬁxed Fatou coordinate ofKg◦ = Dg .
Here is a direct corollary:
Corollary 9.7 The Kleinian lamination Mf is not homeomorphic to Mg . More generally,
there is no pair of hyperbolic and parabolic quadratic maps with topologically the same Kleinian
laminations.

9.4

Quotient tessellations.

Next we focus on how the boundaries of the fundamental regions QPf and QPg are glued bytaking quotient. Since it is determined by the connection of the panels in the quotient, we
introduce the quotient tessellations supported on Sf and Sg . (The quotient tessellations will
also play crucial roles in the next section.) A quotient tile in the lower ends is an equivalentclass [T ] of a tile T in Tessa (f ) or Tessa (g). We set
Tessq (f ) := {[T ] : T ∈ Tessa (f )}Tessq (g) := {[T ] : T ∈ Tessa (g)}.
Recall that the boundaries of tiles in Tessa (f ) and Tessa (g) consist of equipotential, critical,
and degenerating edges. For a quotient tile [T ], we deﬁne its equipotential, critical anddegenerating edges by the equivalent classes of the corresponding edges of T . (Of course
there is no degenerating edges in tiles of Tessq (g).)
Note that quotient panels also make sense. We ﬁrst study how the quotient tiles/panels
in Sf and Sg share their edges. We claim:
Proposition 9.8 (Quotient tiles and their addresses) The quotient tiles have the following properties:(1) The tiles in Tessq (f ) and Tessq (g) tessellates Sf − z f and Sg respectively.
(2) The set z f of degenerating arcs is the union of degenerating edges for all quotient tiles
in Tessq (f ).
(3) Each tile [T ] ⊂ Sf (resp. Sg ) is uniquely represented by a tile T  in the fundamental
region QPf (resp. QPg ). Thus [T ] has a well-deﬁned address given by that of T  .
(4) The map ĥq : Sf − z f → Sg sends the quotient tiles to the quotient tiles without changing
the addresses.
(5) Any tiles in Tessq (f ) share their equipotential or critical edges iﬀ so do the tiles in
Tessq (g) with the same addresses.
Proof. Property (1) comes from the fact that the tiles in Tessa (f ) and Tessa (g) tessellates
Df − Zf and Dg respectively. Since Zf in Df is the union of degenerating edges in thetessellation Tessa (f ), we have Property (2) by taking quotient. Property (3) holds since
QPf and QPg are fundamental regions that are based on the union of panels with angles{θ̂ + } ∪ Ω. To check properties (4) and (5), recall that ĥa : Df − Zf → Df conjugates fˆa andĝa and preserve the address of tiles in Tessa (f ) and Tessa (g). Then (4) and (5) are naturalconclusions.

54

Connection of the edges on Q and P. Let us remark a little more about the connectionof edges. The connection of the equipotential edges are trivial, since they are uniquely glued
in the panels. On the other hand, as indicated on the right of Figure 24, the critical edges
in the fundamental regions QPf and QPg are glued in a complicated way according to the
dynamics. However, by Proposition 9.8(4), the connection of the equipotential or criticaledges are preserved as f → g.
Note that the topological diﬀerence between Sf and Sg is given by the existence of thedegenerating arcs z f (the black thick curves in Figure 24, upper right) for Sf . The degenerating arcs will also play important roles in bifurcation processes, which we will deal with in
the next section.

9.5

Example: The Cauliﬂowers

Here we give a brief summery of our results applied to the Kleinian laminations of theCauliﬂowers (f → g). (The Airplanes can be described in a similar way.)
Recall that the Cauliﬂowers represent the motion of fc with c ∈ (0, 1/4], starting nearf0 (z) = z 2 and terminating at the parabolic f1/4 (z) = g(z) = z 2 + 1/4. For f = fc with
0 < c < 1/4, Corollary 3.4 and Theorem 4.2 imply:(C1) The Kleinian lamination Mf0 is quasi-isometrically equivalent to Mf . In particular,
Mf has a product structure ≈ S0 × [0, 1].
Thus to compare the Klainian laminations of f0 and g, it is enough to investigate thediﬀerence between Mf and Mg . According to the notation in Section 8, we set the principal
leaves f := Λhf /fˆ (a solid torus) and g := Λhg /ĝ (a rank one cusp). By Theorem 8.2,
(C2) There exists a very meek map from the non-principal parts Mf − f onto Mg − g .
(C3) The principal part f and g has the same topology homeomorphic to D × T.
An essential topological diﬀerence appears in the conformal boundaries. Before we deal
with the conformal boundaries, let us remark that we can upgrade (C2) as follows:(C2)’ The non-principal parts Mf − f and Mg − g of the Kleinian laminations are homeomorphic.
Sketch of the proof of (C2)’.
In the dynamics downstairs, every non-cyclic panel
eventually lands on the panel of angle 1/2 and then mapped
to the invariant
 panel of angle
0. It is not diﬃcult to construct a homeomorphism H : ∗=± Πf (1/2, ∗) → ∗=± Πg (1/2, ∗)which extends to the boundaries and agrees with the semiconjugacy h. By using dynamics
and Böttcher coordinate on/outside the Julia set, we can extend the map H to the sphereexcept the union of panels of angle 0 so that H is a conjugacy. This lifts to a conjugacy
between the non-principal part Ĥa : Af − Λaf → Ag − Λag . Then Ĥa also extends to a
conjugacy between Ĥh : Hf − Λhf → Hg − Λhg by [LM, Lemma 9.2] or Lemma 7.2. By taking
quotient we can easily yield a homeomorphism between Mf − f and Mg − g . It extends tothe conformal boundaries since the conjugacy Ĥa preserves the Fatou set and the Julia set.
Conformal boundaries. Let S f and S g denote the upper ends of Mf and Mg , and Sf
and Sg their lower ends. By Proposition 4.1 and Proposition 9.6, we have(C4) The upper ends S f and S g are conformally equivalent to Sullivan’s solenoidal Riemannsurface lamination S0 .
55

(C5) The lower end Sf is compact, but Sg is non-compact.
Principal leaves. Let us check the diﬀerence between f and g in detail. By (C4) the
leafwise upper ends of f and g are conformally the same annulus, say s0 . Let sf and sg
denote their leafwise lower ends in f and g . Then sf is an annulus containing the cyclic−
part Q+f ∪ Qf of the fundamental region. On the other hand, sg consists of two punctured−
disks which has a cuspidal part Q+
g ∪ Qg .
In the sense of the product structure given in Theorem 4.2, we may regard s0 ∪ f ∪ sf
as a product s0 × [0, 1]. However, the product structure obviously breaks for g (Figure 25).
Thus we conclude:(C6) Mf = Mf ∪ ∂Mf has a product structure ≈ S0 × [0, 1], but Mg does not. In particular,
The Kleinian laminations Mf0 and Mg are not homeomorphic.

Figure 25: Caricatures of f and g . Note that the core curve (the dotted circle) of f iscontained in the valley v f = Vf /fˆ. There are Ifh /fˆ-components (not drawn) coiling aroundthe core curve which are sent by ĥq onto geodesics escaping to the cusp (cf. Figure 20).
Remark. By the same argument as in [LM, §6.4], one can show that Mg − g has the
product structure Mg − g  (S0 − s0 ) × [0, 1].

10

Twisting the lower ends

This section is devoted for the bifurcation processes after Case (a) degenerations as in thepreceding section.
Suppose that f = fc varies continuously from one hyperbolic component to another via
parabolic g = fσ . This degeneration and bifurcation process is represented by a couple of Case(a) degeneration pair (f1 → g1 ) and Case (b) degeneration pair (f2 → g2 ) with g1 = g2 = g.
(This cannot happens when g is the root of a primitive copy of the Mandelbrot set, like theCauliﬂowers and the Airplanes.)
The aim of this section is to compare the Kleinian laminations for f1 and f2 . Again byan analogy to the quasi-Fuchsian example (Section 4), we continue to study the topologicaland combinatorial structures of the lower ends. Here is a brief summary of this section:• First we check that the topology of Mfc is leafwise preserved as c moves as above(Proposition 10.1). So we will focus on describing the diﬀerence of the lower ends for
f1 and f2 .

56

• We compare the tessellations Tess(f1 ) and Tess(f2 ). Then we introduce the subdivided
tessellation Tess (f2 ) of Tess(f2 ) so that there is a natural one-to-one correspondence
between the tiles in Tess(f1 ) and Tess (f2 ).
• We deﬁne a fundamental region QP f2 of the action fˆ2a : Df2 → Df2 for the lower
end Sf2 = Df2 /fˆ2a in the same way as we did for f1 . Then we check some common
properties of Sf1 and Sf2 (Proposition 10.2).
• By investigating the connections of the degenerating edges of (subdivided) tiles in thefundamental regions, we will show that the quotient tessellations in Sf1 and Sf2 havediﬀerent combinatorics, given by twisting and dislocation (Theorems 10.4 and 10.5).
• Then we return to the case when f1 is in the main cardioid of the Mandelbrot set andgive a proof of Theorem 1.2.
• Finally as an example, we apply these results to the Rabbits.

10.1

Trans-component partial conjugacies.

Assumption and notation. Let f1 , g and f2 be as above. Let p , q  and l denote as
usual for g = g1 = g2 . To distinguish p, q, l’s of f1 and f2 , we denote them by pi , qi , li forfi (i = 1, 2). Since g is a shared boundary point of two distinct hyperbolic components, we
assume that q  ≥ 2. By Proposition 5.1 we may assume that the Case (a) degeneration pair
(f1 → g1 ) and the Case (b) (f2 → g2 ) satisfy the following:
q1 = q  =: ql1 = l =: lq2 = 1l2 = q  l = ql = ¯l.

andNote that we may regard (f1 → g1 ) as (f → g) in the preceding section satisfying Assumption(a).
Examples. A typical example is given by the Rabbits. More generally, we can take f1 in
the main cardioid and f2 in the center of the p/q-satellite limb.
Note that we always have Θf1 = Θg = Θf2 . For simplicity we denote them by Θ and theircommon invariant lift to T̂ by Θ̂.
To simplify the notation, objects of the form Xfi (i = 1 or 2) will be denoted by Xi if it
does not lead confusion. For example, we write A1 = Af1 , I2 = If2 , etc.
Diﬀerence in the aﬃne and H3 -laminations. Let us roughly compare the topologies
of A1 and A2 . By Theorem 6.6(1), every non-principal leaf of A1 is pinched to be a nonprincipal leaf of Ag , then again plumped to be a non-principal leaf of A2 . In particular, there
is a one-to-one correspondence between the non-principal leaves of A1 and A2 , and one canﬁnd no signiﬁcant topological diﬀerence. On the other hand, each one of the principal leaves
in A2 originally comes from q  principal leaves of A1 . (cf. Figure 15.) Thus the leafwise
topology of the aﬃne lamination changes at the principal part when fc moves continuously
from one hyperbolic component to another via a parabolic map.
One can easily see that the same happens for the H3 -laminations. However, if we turn tothe quotient laminations, the leafwise topology is preserved in the following sense.
Leafwise topology of the quotient 3-laminations.
Let us compare M1 and M2 ,the interiors of the Kleinian laminations. By Theorem 8.2, we have meek pinching mapsĥ1q : M1 − v 1 → Mg and ĥ2q : M2 − v 2 → Mg . This immediately implies:
57

Proposition 10.1 (Leafwise topology) For any non-principal leaf  in Mg − g , let 1 =
−1 

2

1
ĥ−1
1q ( ) and  = ĥ2q ( ) be its preimages, which are non-principal leaves. Then  ,  , and2 are either all isomorphic to solid tori or all isomorphic to H3 .
The principal leaves 1 = f1 , 2 = f2 and g are all homeomorphic. Moreover, g is−1
homeomorphic to the preimages 1 − v 1 = ĥ−1
1q (g ) and 2 − v 2 = ĥ2q (g ).
Hence the inner topology is leafwise preserved when fc moves continuously as above.
Remark. During the motion of fc , the leafwise homotopies are also preserved. It seems
that fc gives a family of faithful representions of the “laminated fundamental group” of M1 .
By Proposition 4.1, the upper ends of these laminations have the same topological structure.
Hence intriguing topological diﬀerences would be observed in the lower ends. Let us compare
the lower ends S1 and S2 of M1 and M2 by a similar method to the previous section.
Partial conjugacies. To investigate the combinatorial change between the lower ends ofthe quotient 3-laminations, we need to compare the tessellations Tessa (f1 ) and Tessa (f2 ). We
start with deﬁning a map which relates the tiles in Tessa (f1 ) to those of Tessa (f2 ).
Let hi : Ĉ → Ĉ be the semiconjugacy from fi (i = 1, 2) to g given in Theorem 5.5. Thenthe restriction hi | C̄ − Ii → C̄ − Igi is a topological conjugacy. Since Ig1 = Ig2 , we have apartial conjugacy κ = h−1
2 ◦ h1 : C̄ − I1 → C̄ − I2 from f1 to f2 ([Ka3, Corollary 4.2]). One
can easily check that κ : C̄ − I1 → C̄ − I2 is lifted to the partial conjugaciesκ̂n : N1 − Iˆ1 → N2 − Iˆ2
κ̂n | An − Iˆ1 → An − Iˆ212

κ̂a : A1 − I1 → A2 − I2 .
In particular, the restriction κ̂a | D1 − Z1 → D2 − Z2 is a conjugacy between fˆ1a |D1 −Z1 andfˆ2a |D2 −Z2 too. Thus by taking quotient we have a homeomorphismκ̂q : S1 − z 1 → S2 − z 2 .
Subdivision of tessellation. By comparing the deﬁnitions of Tess(g1 ) and Tess(g2 ) ([Ka3,§3]), one can easily check thatq−1

Tg2 (θ, m, ∗) =Tg1 (θ, m + µl, ∗)µ=0

for any Tg2 (θ, m, ∗) ∈ Tess(g2 ). Thus Tess(g1 ) is just a subdivision of Tess(g2 ).
Take a tile T1 (θ, m, ∗) ∈ Tess(f1 ). Then there is a homeomorphic image T2 (θ, m, ∗) :=κ(T1 (θ, m, ∗)) in K2◦ = Kf◦2 . We say the family
Tess (f2 ) := {κ(T ) : T ∈ Tess(f1 )}is the subdivided tessellation of K2◦ − I2 . Since Tess(f1 ) and Tess(f2 ) have the same combinatorics as Tess(g1 ) and Tess(g2 ) respectively,q−1

Tf2 (θ, m + µl, ∗)Tf2 (θ, m, ∗) =µ=0

for any Tf2 (θ, µ, ∗) ∈ Tess(f2 ). Now we have a natural tile-to-tile correspondence between
Tess(f1 ), Tess(g1 ) and Tess (f2 ).
58

By Tessa (f2 ) we denote the natural subdivisions of Tessa (f2 ) associated with Tess (f2 ).
More precisely, the subdivided tessellation Tessa (f2 ) is the family of the subdivided tiles
T2a (θ̂, m, ∗) := κ̂a T1a (θ̂, m, ∗) ,where T1a (θ̂, m, ∗) is a tile of Tessa (f1 ). Finally we deﬁne the subdivided quotient tessellation
Tessq (f2 ) as the family of κ̂q ([T ]) with [T ] ∈ Tessq (f1 ).

Figure 26: Tess(f1 ) and Tess (f2 ) for g(z) = z 2 − 3/4. Compare with Figures 2 and 7 in[Ka3].

10.2

Fundamental regions and common properties

Now we are ready to observe the common properties and the combinatorial diﬀerence between
the lower ends S1 = Sf1 and S2 = Sf2 of the quotient 3-laminations by studying the connectionof the quotient tiles.
Fundamental regions.
Recall that we may regard f1 as f in the preceding section(“Assumption (a)”). To compare D1 = Df1 = Df and D2 = Df2 , let us deﬁne the sets in D2
corresponding to Qf (∗) = Q1 (∗) and Pf (∗) = P1 (∗) in the same way. For ∗ = ±, we set
q−1

T2a (θ̂ + , µl, ∗) and P2 (∗) :=Q2 (∗) :=
µ=0


 
Πa2 (ω̂, ∗) − {γ2a (ω̂)}ω̂∈Ω

where we set Πa2 (ω̂, ∗) := κ̂a (Πa1 (ω̂, ∗)), the union of all subdivided tiles of the form T2a (ω̂, µl, ∗)with µ ∈ Z. One can check that the union
QP f2 = QP 2 :=P2 (∗) ∪ Q2 (∗)∗=±

is a fundamental region of S2 = D2 /fˆ2a .
Common properties. For simplicity, we denote the subsets Qi (∗)/fˆia and Pi (∗)/fˆia in Si
by Q∗i and Pi∗ respectively for i = 1 and 2. We ﬁrst check the properties that are common
to f1 and f2 :Proposition 10.2 (Common properties of S1 and S2 ) The lower ends S1 = D1 /fˆ1aand S2 = D2 /fˆ2a of M1 and M2 have the following common properties:(1) The (subdivided) quotient tiles in Tessq (f1 ) and Tessq (f2 ) tessellate S1 − z 1 and S2 − z 2
respectively. In particular, the sets z 1 and z 2 of degeneration arcs are the unions of thedegenerating edges in these (subdivided) quotient tessellations.
59

(2) The critical and equipotential edges of the quotient tiles in S1 = D1 /fˆ1a have the samecombinatorics as those of the subdivided quotient tiles in S2 = D2 /fˆ2a .
(3) For i = 1, 2 and ∗ = ±, we have Q∗i ∪ Pi∗ = Pi∗ and Pi+ ∪ Pi− = Si . In particular, every
quotient panel [Πai (ω̂, ∗)] in Pi∗ accumulates on Q∗i .
(4) Both S1 and S2 are compact.
Proof. Property (1) is obvious by the deﬁnition of the quotient tessellations. Property (2)holds since the critical and equipotential edges of tiles in Tessa (f2 ) have the same combinatorics as those of Tessa (g1 ) and Tessa (f1 ). For (3) and (4), one can just apply the sameargument as Propositions 9.5 and 9.6 to f2 .
So we may observe the main diﬀerence between S1 and S2 in the connection of thedegenerating edges.

10.3

Combinatorial diﬀerence along degenerating edges

Let us investigate how the quotient tiles are glued along the degenerating edges in the lower
ends S1 and S2 .
Degenerating arcs in the lower ends. By Proposition 10.2(1), the degenerating edges lie
on the set of degenerating arcs z i = Zi /fˆia (i = 1, 2). As we did for the fundamental domainsof the lower ends, it is convenient to introduce the cyclic and non-cyclic decompositions of
the degenerating arcs z 1 and z 2 . We claim:
Proposition 10.3 (Decomposition z i = xi  y i ) For each i = 1 or 2, there exists only
one closed z i -component given by
xi := (Di ∩ IOi )/fˆia−+−which is contained in Q+
i ∩ Qi . The complement y i := z i − xi is contained in Pi ∩ Pi .
Moreover, each path-connected component of y i (y i -component for short) accumulates on xi .

Proof. Let us to go back to the dynamics fˆia : Di → Di . Recall that Zi = Di ∩ Ii . Now
the union of cyclic Zi -components is given by Xi := Di ∩ IOi . By deﬁnition of the closed set
Qi (+) (resp. Qi (−)), any point in Xi lands on the degenerating edges of Qi (+) (resp. Qi (−))
just once (or twice on the endpoints) by the dynamics of fˆia . Thus the quotient xi := Xi /fˆia−
is contained in the boundaries of both Q+
i and Qi . In particular, it is a simple closed curve.
Since the set Yi := Zi − Xi is the union of non-cyclic Zi -components, its quotient y i :=z i − xi consists of z i -components that are homeomorphic to a path-connected component of
Dfi ∩ Ifi downstairs. Hence any z i -components except xi cannot be a closed curve. Since
non-cyclic Zi -components are contained in non-cyclic panels in Di , the set y i is contained in
the non-cyclic part Pi+ ∩ Pi− in the lower end.
Since any panel in Pi∗ accumulates on Q∗i (Proposition 9.5), any y i -component accumulates on xi .
−In the following we consider the connection of the quotient tiles along xi in Q+
i ∩ Qi and+−y i in Pi ∩ Pi separately. The question is:+
For each i = 1 or 2, when do (subdivided) tiles [T + ] ∈ Q+i (resp. Pi ) and−−−[T ] ∈ Qi (resp. Pi ) share their degenerating edges?

60

The answer will give us a particular diﬀerence between S1 and S2 .
Diﬀerence in the cyclic parts Q˜i . By deﬁnition, the sets Q1 (∗) and Q2 (∗) consist ofthe (subdivided) tiles T1a (θ̂ + , µl, ∗) and T2a (θ̂ + , µl, ∗) with 0 ≤ µ < q. We ﬁrst claim:
Theorem 10.4 (Twisting Q˜i ) In Si , the tiles in Q∗i are glued along the degenerating edge
xi := Xi /fˆi with the following combinatorics:a + (a) The degenerating edge of the quotient tile [T1a (θ̂ + , µl, +)] in Q+
1 is glued to that of [T1 (θ̂ , µ l, −)]−in Q1 iﬀ µ = µ .

(b) The degenerating edge of the subdivided quotient tile [T2a (θ̂ + , µl, +)] in Q+
2 is glued to
−
a
+


that of [T2 (θ̂ , µ l, −)] in Q2 iﬀ µ ≡ µ + p̃ for some p̃ with p̃ p ≡ 1 modulo q.
Proof. The statement for i = 1 is clear by deﬁnition of Q1 (∗). Let L2 be the principal leafthat contains the external ray of angle θ̂ + . Then L2 also contains both Q2 (+) and Q2 (−).
The cyclic Zi -component in L2 is given by the non-compact star-like graph L2 ∩ IO2 with
its central joint contained in J2 removed. Thus it has q connected components I0 , . . . , Iq−1
in counterclockwise order. Now L2 ∩ IO2 divides L2 in q sectors and the tiles in both Q2 (+)
and Q2 (−) are contained in one of these sectors that contains the external ray of angle θ̂ + .
We may assume that I0 and I1 bounds this sector. Then the degenerating edge of tiles in
Q2 (+) and Q2 (−) are contained in I1 and I0 respectively. (The right square of Figure 27
shows this situation in L2 for p/q = 1/3. The left square shows the corresponding picture for
i = 1.)
By the action of fˆ2l : L2 → L2 , the q periodic external rays in L2 and I0 , . . . , Iq−1 arepermuted with rotation number p/q. (cf. [Ka3, Propositions 2.2 and 2.3].) In particular, wehave fˆ2lk (Ij ) = Ij+kp taking subscript modulo q. To glue the degenerating edges of Q2 (+)
and Q2 (−), it is necessary for I0 to be mapped to I1 by some fˆ2lk . Such a k is given by theequation 0 + kp ≡ 1 modulo q. This k is our p̃ in the statement.


Figure 27: For i = 1 (left) and i = 2 (right), the sets Qi (+) and Qi (−) are subsets of theprincipal leaves containing the external ray of angle θ̂ + (the blue dotted lines). The thickened
lines indicate the degenerating arcs.
Dehn twist along x1 . The theorem above indicates what is the diﬀerence between the
−
+
−
+
−
−
cyclic parts Q+
1 ∪ Q1 and Q2 ∪ Q2 : First cut Q1 ∪ Q1 along x1 , and then twist Q1 by(combinatorial) angle p̃/q ∈ T and glue them again. Then we have combinatorially the same−
object as Q+
2 ∪ Q2 with the seam x2 . We say this operation a combinatorial p̃/q-Dehn twist

61

−
of Q+
1 ∪ Q1 . In the next paragraph we will see that p̃ is more speciﬁed when we look at thischange in Pi+ ∪ Pi− .

Diﬀerence in the non-cyclic parts Pi˜ . Next we consider Pi± in the lower end Si for i = 1
and 2. Let T + ∈ Pi (+) and T − ∈ Pi (−) be the tiles of addresses (ω̂, µl, +) and (ω̂  , µ l, −)respectively. (If i = 2 addresses are taken as tiles in Tessa (f2 ).) As in the question above, we
assume that the quotient tile [T + ] ∈ Pi+ and [T − ] ∈ Pi− share their degenerating edges. Then
there exists a unique (subdivided) tile U − ⊂ Di of address (θ̂, µl, −) that actually shares its
degenerating edge with T + in the aﬃne lamination before taking the quotient. For ﬁxed i = 1
or 2, the quotient tiles [T + ] and [T − ] share their degenerating edges in Si iﬀ there exists a
unique integer p̃ such that fˆip̃l (T − ) = U − . The following theorem says that such a p̃ depends
only on i and p/q:
Theorem 10.5 (Dislocating Pi ) For ﬁxed i = 1 or 2, let T + , T − and U − be the (subdivided) tiles in Pi (+) ∪ Pi (−) deﬁned as above. Then the integer p̃ with fˆp̃l (T − ) = U − isdetermined as follows:(a) If i = 1, then T − = U − and thus p̃ = 0. In other words, the quotient [T + ] shares its
degenerating edge with [T − ] iﬀ ω̂ = θ̂ = ω̂  and µ = µ .
(b) If i = 2, then p̃ is uniquely determined so that |p̃| < q/2 and p̃ p ≡ 1 modulo q. In otherwords, the quotient tile [T + ] shares its degenerating edge with [T − ] iﬀ θ̂ = δ̂p̃l (ω̂  ) andµ = µ + p̃.
In particular, p̃ does not depend on the choice of ω̂ and µ.
−
Note that this theorem is consistent with p̃/q-Dehn twist of Q+
1 ∪Q1 described in Theorem
10.4.

Proof. By Θ0 we denote the type of γ2 (θ0+ ) (or β0 ); that is, the set of q angles which is cyclic
under the iteration of δl and contains θ0± . For ω̂ = (ω−n )n≥0 ∈ Ω, there exists a maximal
N = N (ω̂) such that ω−N l ∈ Θ0 . Since ω0 = θ0+ and ω−n = θ0+ for all n > 0 , we have0 ≤ N < q. In particular, the q angles
ω−N l , ω(−N +1)l , . . . , ω0 = θ0+ , . . . , ω(−N +q−1)l(where ωk := δk (ω0 ) when k > 0) coincide with the angles in Θ0 .
Now T + ∈ Pi (+) and T − ∈ Pi (−) have addresses (ω̂, µl, +) and (ω̂  , µ l, −) respectively.
Set N := N (ω̂) and N  := N (ω̂  ). Since U − shares its degenerating edge with T + in Di
before taking the quotient, the address of U − is of the form (θ̂, µl, −) with non-cyclic θ̂ =(θ̂−n )n≥0 ∈ Θ̂. Note that by the dynamics downstairs tiles of addresses (ω−n , µl − n, +) and(θ−n , µl − n, −) share their degenerating edges for all n ≥ 0. Moreover, we have/ Θ0 for all n > N l; and• θ−N l ∈ Θ0 and that θ−n ∈• θ0 = θ0+ or θ0 = θ0− according to i = 1 or 2.
In the case of i = 1, the tiles of addresses (ω−n , µl − n, +) and (ω−n , µl − n, −) share theirdegenerating edges so we have ω−n = θ−n for all n ≥ 0. Hence the statement is clear for
i = 1.
Suppose that i = 2. Since fˆ2p̃l (T − ) = U − implies δ̂p̃l (ω̂  ) = θ̂, we have δp̃l (ω0 ) = θ0 and−N  + p̃ = −N . Recall that ω0 = θ0+ and θ0 = θ0− . By the same argument as Theorem 10.4,
such a p̃ must satisfy p̃ p ≡ 1 mod q. Since q angles
+



ω−N
 l , . . . , ω0 = θ0 , . . . , ω(−N  +q−1)l62

coincide with Θ0 , they contain just one angle equal to θ0− . Hence we have −N  ≤ p̃ ≤−N  + q − 1. Since N  = p̃ + N and 0 ≤ N < q, we have −q/2 < p̃ < q/2. Now p̃ is uniquelydetermined and depends neither on ω̂ nor µ.
The combinatorial p̃/q-Dehn twist of the lower ends. Proposition 10.2(1), Theorems
10.4 and 10.5 above show that the combinatorial diﬀerence between S1 and S2 is causedby the connection of the degenerating edges. Let us observe it in a continuous motion of
the parameter c of f = fc . Let c move from one hyperbolic component H1 to one of its
satellite component H2 via the parabolic parameter σ. (That is, σ is the root of H2 .) Our
degeneration pair (fi → g = fσ ) represents the motion within Hi for each i = 1 and 2.

Figure 28: Dislocation of Pi along twisted Qi . This caricature only illustrates the fact thatthe panels in P2∗ are slided by p̃l tiles along y 1 . They are also shuﬄed, i.e., diﬀerent pairs ofpanels are glued along y 1 and y 2 .
As we have seen in Proposition 9.5 and Proposition 10.2, the sets Pg∗ and Pi∗ accumulate
on Q∗g and Q∗i for all i = 1, 2 and ∗ = ± respectively. Their boundaries are glued in exactly
the same way on the critical edges, but there is no edges in Q∗g ∪ Pg∗ corresponding to thethe degenerating edges in Q∗g ∪ Pg∗ (Figure 24). More precisely, when c is in H1 ﬁrst, thedegenerating edges z 1 form a “laminated path” in S1 which has only one closed path x1 withthe other y 1 -components winding around (Proposition 10.3). They are pinched and pushed
to “inﬁnity” as f1 → g (c → σ) then S1 looses compactness at Sg (Proposition 9.6).
Next when c moves into H2 , the degenerating edges z 2 = Z2 /fˆ2 are again plumped up
but the connection of the degenerating edges are changed as Theorems 10.4 and 10.5 above
indicate. The diﬀerence is the combinatorial p̃/q-Dehn twist along x1 , and the combinatorial
dislocation by p̃l (subdivided) tiles along y 1 -components. In particular, the combinationof the glued panels along y 1 are shuﬄed according to the relation θ̂ = δ̂p̃l (ω  ) in Theorem
10.5(b). Such p̃ is uniquely determined by p/q and these two operations are totally consistent
with each other. So we may consider that the union of these operation is a natural twisting
operation of the lamination along z 1 when c moves from H1 to H2 . We call this operation
the combinatorial p̃/q-Dehn twist of the lower ends.

10.4

Topological diﬀerence of the Kleinian laminations

The phenomenon above looks similar to the example of quasi-Fuchsian groups in Section 4,
but there is a particular diﬀerence also: The total topology of the Kleinian lamination may
change.
In the quasi-Fuchsian example, the Kleinian manifold Mt has the same product structure≈ S × [0, 1] for all 0 < |t| < 1. (See [MT, Theorem 4.34] for example). This fact corresponds
to:63

Theorem 10.6 (Product structure) Let f = fc be a quadratic map. Then the Kleinian
lamination Mf is homeomorphic to the product S0 × [0, 1] if and only if c is in the main
cardioid of the Mandelbrot set.
Proof. By Theorem 4.2, the Kleinian lamination Mf has a product structure S0 × [0, 1]
when c is in the main cardioid.
Conversely, suppose that there exists a topological homeomorphism H : Mf → S0 ×
[0, 1]. Then the conformal boundary ∂Mf must consist of two indecomposable connectedcomponents.
If the Fatou set has only one connected component (= Bf , the basin at inﬁnity) downstairs, then ∂Mf = Ff /fˆ cannot have two components. Thus the interior Df of the ﬁlled
Julia set Kf is not empty.
Let γ = γf (0) be the β-ﬁxed point of f . Then the leaf  = γ of Mf associated withthis repelling ﬁxed point is a solid torus. The leafwise conformal boundary ∂ has two
indecomposable components corresponding to Bf and Df , the upper and lower ends. Since
we have the homeomorphism H, the image H() must be a solid torus and the conformalboundary of H() must be two annuli. This can happen only if c is in the main cardioid,
otherwise the lower end of  has countably many path-connected components or c = 1/4.
When c = 1/4, the lower end Sf is non-compact by Theorem 9.6. This contradicts the factthat S0 is compact, and that H sends the boundary component to the boundary component.
This yields Theorem 1.2 in the introduction. By using this theorem, we immediately havethe following example:
Corollary 10.7 Let f1 be in the main cardioid, and f2 in the p/q-limb of the Mandelbrot
set. Then the Kleinian lamination M1 is not homeomorphic to M2 .
Now it is natural to ask whether we can generalize this corollary or not:Question. For any pair of Case (a) degeneration pair (f1 → g) and Case (b) (f2 → g),
can the Kleinian lamination M1 be homeomorphic to M2 ?
It suﬃces to show that the lower end S1 cannot be homeomorphic to S2 . However, theauthor do not have the answer even for the following question:
Question. Can Sullivan’s lamination S0 be homeomorphic to the lower end Sf of some
hyperbolic quadratic map f = fc with c outside the main cardioid?

10.5

Example: The Rabbits

To conclude this paper let us consider a typical degeneration and bifurcation process when
the parameter c moves from 0 to the center of the 1/3-limb (Douady’s rabbit, with the value
denoted by crab ) of the Mandelbrot set. (cf. Figure 1 of Part I [Ka3].)
Let (f1 → g) be the Case (a) Rabbits and (f2 → g) the Case (b) Rabbits. Set f0 (z) = z 2
and f3 (z) = z 2 + crab . By Corollary 3.4 and Theorem 4.2, we have:(R1) The Kleinian lamination Mf0 (resp. Mf3 ) is quasi-isometrically equivalent to Mf1(resp. Mf2 ). In particular, Mf1 has a product structure ≈ S0 × [0, 1].
Thus it is enough to check the changes of laminations associated with (f1 → g) and(f2 → g) to compare the Kleinian laminations of f0 (z) = z 2 and Douady’s rabbit.

64Quotient 3-laminations. The principal leaf 1 = Λh1 /fˆ1 (a solid torus) is pinched to be
the principal leaf g = Λhg /ĝ (a rank one cusp), then blown up to become the principal leaf
2 = Λh2 /fˆ2 (again a solid torus). By Theorem 8.2 we have:(R2) For each i = 1 or 2, there exists a very meek map ĥiq : Mi − i → Mg − g .
(R3) The principal part 1 , g , and 2 have the same topology homeomorphic to D × T.
Thus we may conclude: During the degeneration-and-bifurcation process from f0 to f3 via
f1 , g and f2 , all the leaves of the quotient 3-lamination are topologically preserved.
In addition, let us remark that the same argument as the proof of (C2)’ yields an upgradeof (R2):(R2)’ The non-principal part M2 − 2 is homeomorphic to Mg − g .
Conformal boundaries. We may ﬁnd more signiﬁcant diﬀerence in the conformal boundaries. An intriguing fact is, though this motion of the parameter passes across the hyperbolic
components, the situation is still analogical to the quasi-Fuchsian deformation.
For i = 1 or 2, let S i and S g (resp. Si and Sg ) be the upper (resp. lower) ends of Mi
and Mg . By Propositions 4.1, 9.6, and 10.2, we have(R4) The upper ends S i (i = 1, 2) and S g are all conformally homeomorphic to Sullivan’ssolenoidal Riemann surface lamination S0 .
(R5) The lower ends S1 and S2 are compact, but Sg is non-compact.
Product structure. Next we check the product structures of Mi (i = 1, 2) and Mg . By
Theorem 10.6 we have:(R6) The quotient 3-lamination with conformal boundary M1 has a product structure ≈S0 × [0, 1], but Mg and M2 do not.
See Figure 29. By summing up (R5) and (R6) (or by Corollaries 9.7 and 10.7), we have:(R7) The Kleinian laminations M1 , Mg , and M2 are not homeomorphic each other.
In particular, we have:(R7)’ The Kleinian lamination of f0 (z) = z 2 is not homeomorphic to that of Douady’s rabbit.
Now Theorem 1.1 in the introduction can be checked by (C6), (R1), and (R7).
Dehn twist in the lower ends. The most intriguing part is the combinatorial diﬀerence
between S1 and S2 given by the twisting (and sliding) operation along z i = Zi /fˆi .
By the argument of Sections 9 and this section, S1 and S2 have the cyclic and non-cyclic
decompositionQ∗i ∪Pi∗ (i = 1, 2).
Si =
∗=±

∗=±

As f1 → g, the union of the quotient degenerating arcs z 1 is pinched and pushed away to“inﬁnity”. As fc moves from g to f2 , the set of quotient degenerating arcs appears again as z 2
but the connection of (subdivided) tiles along z 1 and z 2 are diﬀerent. By applying Theorem
10.4 and Theorem 10.5 to our case of p/q = 1/3 and l = 1, we have p̃ = 1 and conclude:(R8) During the degeneration and bifurcation process from f1 to f2 via g, z 1 is pinched andthen plumped with combinatorial p̃/q = 1/3-Dehn twist to become z 2 .
65

See Figures 24 and 28 again.
Principal leaves. Let us observe (R8) in the principal leaves 1 , g , and 2 with conformalboundaries. Their interiors are topologically the same by (R3). Moreover, by (R4) their leafwise upper ends are conformally the same annuli, say s0 . (Note that this s0 is diﬀerent froms0 in the Cauliﬂowers: the modulus is three times larger.) Hence the topological diﬀerence
is concentrated in the lower ends.
Let s1 , sg , and s2 denote the leafwise lower ends of 1 , g , and 2 respectively. As shown in
the left of Figure 29, s1 is an annulus and it has countably many z 1 -components (black thick−curves). It also contains the cyclic part Q+
1 ∪ Q1 of the fundamental region QP1 . As f1 → g,these z 1 -components are pinched and we have non-compact lower end sg , which consists ofcountably many path-connected components. In particular, there are two of them containing−Q+
g and Qg , the cuspidal parts. Then g is perturbed to be f2 and the z 2 -components appear.
The z 1 and z 2 consist of the degenerating edges of tiles in Tessq (f1 ) and Tessq (f2 ), but theconnection of these edges are diﬀerent as described in (R8).

Figure 29: Caricatures of 1 , g , and 2 from left to right.
Remark: Rabbits in rabbit, etc.
The similar change happens when c moves fromc3 = crab to another hyperbolic component H  via another parabolic parameter. Mainly the
topology of the quotient 3-lamination is leafwise preserved, but the lower end changes its
combinatorial structure when it goes into H  . The location and rotation number of Dehn
twist changes according to the angles of the landing ray at the root of H  .
Now it is natural to consider what may happen after inﬁnitely many bifurcations via
parabolic parameters. This type of bifurcations are described in terms of renormalization.
At least for the aﬃne laminations of inﬁnitely renormalizable quadratic maps with a priori
bounds, we already have a reasonable picture. See [CK].

AAppendix: Remarks on Hénon mappings

For hyperbolic fc (z) = z 2 + c and a ∈ C∗ , we deﬁne the Hénon map Hc,a : C2 → C2 by
 


x
fc (x) − ay
→ Hc,a(z) =.
Hc,a : z =yxThis map has a constant Jacobian a thus invertible. When |a|  1, wemay regard Hc,a as an (z) is bounded
2-dimensional perturbation of fc (z) = z 2 + c. We say z ∈ K± (Hc,a ) if Hc,a
as n → ±∞. We set J± (Hc,a ) := ∂K± (Hc,a ), the forward and backward Julia sets. They are
all invariant under the dynamics of Hc,a.
In [HO], Hubbard and Oberste-Vorth showed the following:
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Theorem A.1 (Theorem 5.1 of [HO]) If |a|  1, there exists a homeomorphism Φ− :
ˆ → J− (Hc,a ) which conjugates fˆc and Hc,a .
Nfc − {∞}
ˆ is the natural extension of f : C → C. Hence J− (Hc,a ) minus theNote that Nfc − {∞}attracting cycle of Hc,a is a (topological) realization of the aﬃne part Anfc and the aﬃnelamination Afc in C2 . (Recall that Anf and Af are identiﬁed when f is hyperbolic.) Thismeans that our investigation on natural extensions and aﬃne parts in Section 6 is also an
investigation on J− of some Hénon maps.
For example, take two distinct degeneration pairs (f1 → g) and (f2 → g) such that ci offi (z) = z 2 + ci are contained in the distinct hyperbolic components for i = 1 and 2. Then
there exists a suﬃciently small a = 0 such that the Hénon maps H1 = Hc1 ,a and H2 = Hc2 ,ahave properties as in the theorem above. The maps H1 and H2 can be arbitrarily close in
the parameter space, but they have diﬀerent dynamics on J− by our investigation on aﬃneparts and aﬃne laminations in Section 6. In particular, both J− (H1 ) and J− (H2 ) support
tessellation induced by Tessn (f1 ) and Tessn (f2 ) respectively.
Here we give a brief application:
Theorem A.2 (Distinct J ` .) The backward Julia sets J− (H1 ) and J− (H2 ) are not homeomorphic.
Proof. By the theorem above, J− (Hi ) is homeomorphic to Nfi − {∞}. The irregular pointsof Nfi −{∞} are the cyclic lift of the attracting cycle of fi . Since f1 and f2 have diﬀerent number of attracting points, the sets Nf1 − {∞} and Nf2 − {∞} cannot be homeomorphic. (Notethat an irregular point is characterized as a point that cannot have a laminar neighborhoodwith product structure.)Stable and unstable manifolds. Conversely, one can translate some notion for the Hénonmappings to our aﬃne laminations. For example, we can ﬁnd “stable and unstable manifolds”
in the aﬃne laminations. Take a repelling ﬁxed point z of f and let ẑ be its lift in Anf . Thenthe action of fˆ : L(ẑ) → L(ẑ) is conjugate to an aﬃne expanding map on C. We may
consider it the “unstable manifold” of ẑ. On the other hand, there is a ﬁber Tz := π −1 (z) of
z invariant under fˆ. One can easily check that the action fˆ : Tz → Tz is contracting in thenatural topology. In particular, we have fˆn (ζ̂) → ẑ for any ζ̂ ∈ Tz . We may consider thisﬁber direction the “stable manifold” of ẑ.
The notion of “stable and unstable manifolds” in the aﬃne laminations are not only for
quadratic maps. More generally, it may be intriguing to consider the following problem: Fora rational map f of degree ≥ 2, formulate the stable and unstable manifolds in the universal
dynamics fˆu : Û a → Û a . See also paragraph 10 of [LM, §10] and Appendix of [BS].

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