数理解析研究所講究録
第 1571 巻 2007 年 155-171

155

Twisting operations in Lyubich-Minsky laminations
associated with bifurcations of quadratic maps
Tomoki Kawahira (川平友規)
Graduate School of Mathematics
Nagoya University

Abstract
This note gives a brief introduction to Lyubich and Minsky’s hyperbolic 3laminations associated with hyperbolic and parabolic quadratic maps. We will
see that a twisting operation naturally appears when a quadratic map bifurcates.
This note is based on my talk at RIMS, Kyot , on 7 December 2006.
$0$

1

Introduction: Sullivan’s dictionary

1.1

Complex Dynamics

Iteration theory of rational functions.
deg $f\geq 2$ . For, we consider its orbit

Let $f$ :$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$be a rational map of$z_{0}\in\overline{\mathbb{C}}$$\overline{\mathbb{C}}\ni z_{0}z_{1}z_{2}\underline{f}\underline{f}\underline{f}\ldots$

When we perturb the initial value.

a little, the orbit may change slightly (stably), or
drastically (unstably). To distinguish the properties of orbits, we define the Fatou set:
$F_{f}:=$$z_{0}$(The maximal open set where$\{f^{n}\}_{n\in N}$

is normal);which is the domain of stability. (Whereis the times iteration of ). Indeed, we
can replace “normal” by “equicontinuous” in the definition. Thus the orbits startingat the Fatou set are stable under perturbation. Its complement
gives
the unstable orbits. We callthe Julia set, which is the chaotic locus. Note thatandare invariant sets of.
$J_{f}=S^{1}$ and
Example. $f(z)=z^{2}$. In fact,or compact
uniformly on
as $narrow 0$ . By taking a semiconjugacy of by $w=\phi(z)=z+z^{-1}$ ,one can check that $g(w)=w^{2}-2$ has [-2, 2] as its Julia set.
$f^{n}$$f$$n$$J_{f}:=\overline{\mathbb{C}}-F_{f}$$F_{f}$$J_{f}$$f^{\pm 1}$$J_{f}$$\Rightarrow$$F_{f}$$f^{n}arrow 0$$F_{f}=\overline{\mathbb{C}}-S^{1}$$f$$\infty$

156

1.2

Sullivan’s Dictionary

In early $80’ s$ , D.Sullivan introduced a dictionary between (finitely generated) Kleiniangroups and complex dynamics (iteration theory of rational maps). Here are some ofthe entries:

f.g. Kleinian gr.
Domain of disconti.
Limit set
Fixed points dense in
Ahlfors finiteness thm.

$\tau_{t}\sim \mathbb{C}$

Rational map
Fatou set
Julia set
Periodic points dense in
No wandering $doma\dot{i}$ thm.
$f\cap \mathbb{C}$$F_{f}$$\Omega_{\Gamma}$$J_{f}$$\Lambda_{\Gamma}$$\Lambda_{\Gamma}$$J_{f}$

Sullivan gave proofs of the two theorems below in the table by using a similar methodof the quasiconformal Teichm\"uller theory. Now our slogan is: Share methods in boththeories! However, there still are missing entries:
expanding
Structurally stable
field
on
line
invariantNo
Poincar\’e ext.

7 ( $\Leftarrow is$ known.)$\Rightarrow$?

$\Lambda_{I^{\backslash }}’$$\Gamma r\backslash \mathbb{H}^{3}$

Hvperbolic 3-manifold

No $conforma1ext???$ .

$f\cap \mathbb{H}^{3}$$M_{\Gamma}$

The ffist two ? are affirmatively conjectured and they imply abig result on densityof hyperbolic rational maps. To get anew $ins$ ight for these conjecturae, it would beof aKleinian group
nice for complex dynamics to have ageometric realization like. So far, the only ctdidate for “???” below is Lyubich and Minsky’s hyperbolic Sintroduced in [LM]. They proved arigidity thmrem of acertain classlamination
Marden’sof rational maps by an analogous argument to Mostow’s rigidity theorem
isomorphism theorem.
Outline of this note. In Section 2, we roughly summerize properties of quadratic
maps with connected Julia sets. Section 3is devoted for arough sketch of the construction of Lyubich-Minsky laminations. Section 4gives aworked out example of the
Lyubii-Minsky hyperbolic 3-laminations and their degeneration. This note is basedon some hmdamental facts on complex dynamics. Readers may refer [Mi].
This research is partially supported by Inamori Foundation
Acknowledgement.
and JSPS.
$s$$M_{\Gamma}$$\Gamma$$\mathcal{M}_{f}$$\bm{t}d$2Dynamics of quadratic maps

2.1

Quadratic dynamics and the Mandelbrot set. This is
Let us consider quadratic maps of the form $f(z)=f_{c}(z)=z^{2}+c$.
ofenough general, since any quadratic function is affinely conjugate to one such$(c\in \mathbb{C})$$f_{c}’ s$157

Basin at infinity. Since $f(z)=z^{2}(1+c/z^{2})\approx z^{2}$ near
We define the basin at infinity by$B_{f}$$;=\{z\in\overline{\mathbb{C}} :$\infty,$$\infty$

attracts nearby points.

f^{n}(z)arrow\infty\}\subset F_{f}$which is an open neighborhood of . Note that$f^{\pm 1}(B_{f})=B_{f}$ . Here is a classical result on
and :$B_{j}$$\infty$$B_{f}$

is completely invariant, i.e.,$J_{f}$

Theorem 2.1 (Fatou, Julia) For any $f=f_{c}$ , the boundary of
Moreover, either
1.

$B_{f}\cong D\Leftrightarrow J_{f}$is connected; $or$2.

$B_{f}\not\cong D\Leftrightarrow J_{f}$is a Cantor set.

$B_{f}$

coincide utth

$J_{f}$.

We define the Mandelbrot set by$M$

Now suppose that $B_{f}\cong D$$:=\{c\in \mathbb{C} :$\Leftrightarrow$$c\in$

B_{f}\cong D\}$.

M. Then it is known that there exist a$s$

Figure 1: The Mandelbrot set with its boundary emphasized.

and $fr\vee B_{f}$:between the actions
unique conformal conjugacy
wesatisfying $\Phi_{f}(z)/zarrow 1(zarrow\infty)$ (Figure 2). In other words, the dynan ‘csthroughacting on:
observe is the image of the dynamics of$f_{0}\cap
B_{fo}$ ,dynamics. This is a conformal deformation of thea $co$ nformal lens”$\Phi_{f}$$f_{0}r\sim B_{f_{0}}$$B_{f_{0}}arrow B_{f}$$f \bigcap_{-}B_{f}$$f_{0}$$zrightarrow z^{2}$$B_{fo}=\overline{\mathbb{C}}-D$$\Phi_{f}$which gives a quite similar situation to Bers’ simultaneous uniformization.
Here is another entry of the dictionary:Wheregroup.

$Q\mathcal{F}$is the quasi-Fuchsian space, a deformation space of a hyperbolic surface

158

Figure 2: Conformal deformation of the dynamics in

2.2$B_{f_{0}}$.

Degeneration v.s. Bifurcation

Let us change the parameter of $f=f_{c}$ : $z\mapsto z^{2}+c$ from $c=0$ (small white circle
below) as in the curve of Figure 3. It is convenient to define the filled Julia set by$c$Figure 3: Rabbits. Note that

. In Figure 3, the three triangles indicate thewith their filled Julia sets (with some “equipotential curves”) drawn on
the right. The first one from below, say, has an attracting fixed point and its
Julia set
is a Jordan curve. The interior of the filled Julia set
is colored in gray.
The next one, say
has a fixed point with multiplicity in the following sense:
It is a simple root of the equation $g(z)=z$ , but it is a multiple root of $g^{3}(z)=z$ (in$K_{f}$$:=\overline{\mathbb{C}}-B_{f}$parameter$\partial K_{f}=J_{f}$$c$$f_{1}:=f_{c_{1}}$$J_{f\iota}$$K_{f_{1}}$$g=f_{\sigma)}$$\beta$general this equation gives periodic points of period 3) with multiplicity 4. Hence any
perturbation of this causes a bifurcation of this multiple periodic points. The Julia
$g$

159

set
is no more a Jordan curve, andhas countably many connected components.
The third one, say, has periodic points of period three that attract nearby
points (the attracting cycle). It is known thatis homeomorphic to .
The key to describe the difference between
is the degeneration andbifurcation process with respect to .
has periodic points of period 3 that repel
nearby points (a repelling cycle). As the parameter moves &om to , the repelling
cycle and the attracting cycle of merge into one point, . As the parameter moves
from to , the degenerated periodic points bifurcate again to be the attracting cycle
and the repelling fixed point. Figure 4 shows this process with their nearby dynamics.
$J_{g}$$K_{\mathring{g}}$$f_{2}=f_{c_{2}}$$J_{g}$$J_{f_{2}}$$f_{1}^{3},$$\beta$$g^{3},$$f_{2}^{3}$$f_{1}$$c$$\sigma$$\sigma$$c_{1}$$f_{1}$$\beta$$c_{2}$$\backslash _{\backslash }\subset A^{X}\supset\swarrow o_{\dagger}^{\dagger_{\nwarrow)}\backslash }o\dagger o_{\nearrow}$$C_{r\backslash }^{\backslash _{\bullet\supset}}\backslash \},\swarrow C^{\downarrow}\nwarrow$$\subset\nearrow C_{\dagger}^{\backslash }\backslash \{\prime Y_{q\backslash }^{\supset}$$\bullet$attracting$\bullet$parabo 1 ic

Figure 4: Dynamics of$O$$f_{1}^{3},$$g^{3},$

repal I ing$f_{2}^{3}$

Let us make the terninoloy more precise: For aperiodic point of period , we
say $(f^{l})’(z)$ is the multiplier of . We $ako$ say that is auracting if $|(f^{t})’(z)|<1$ ,repelling if $|(f^{l})’(z)|>1$ , and indifferent if $|(f^{l})’(z)|=1$ . In particular, we say isparabolic if $(f^{l})’(z)$ is aroot of unity, or equivalently, is amultiple root of $f^{n}(z)=z$
for some . The map $f_{c}(z)=z^{2}+c$ (or the parameter c)is called $hyperbolic/parabolic$(parameter) if has $\bm{t}attracting/parabolic$ cycle.
By the Douady-Hubbard theory [DH], hyperbolic parameters are deternined bythe (combinatorics of the attracting cycles and its multiplier valued in D. Indeed,is isomorphic
each connected component of the set of hyperbolic parameters
to D. The boundaries of two of such components may share at most one parabolic
parameter. When moves $hom$ one component of $H$ to tother, apair of degenerationbiMcation process as in Figure 3happens with wrious multiplicity of periodic
points depending on the parabolic parameter between the two components.
In the following, we try to observe this hyperbolic-to-parabolic degenemtion prooessand $parabolicarrow to$ -hyperbolic bifurcation process in tems of Lyubich-Minsky laminations.
Now let us go to the definition of the Lyubich-Minsky laninations.
$z$

$z$

$l$

$z$

$z$

$z$

$n$

$f_{c}$$H\subset M^{o}$$c$$\bm{t}d$160

Lyubich-Minsky laminations

3

Here we roughly sketch the construction of the Lyubich-Minsky laminations associatedwith hyperbolic and quadratic maps. Assume: $f=f_{c}$ : $z\mapsto z^{2}+c$, hyperbolic orparabolic. The construction breaks into four steps:
Stepl. To get an invertible dynamics, we construct the inverse limmit.
with lifted action$\mathcal{N}_{f}=\lim_{arrow}(f,\overline{\mathbb{C}})$of$f\cap \mathcal{N}_{f}$$f\cap\overline{\mathbb{C}}$Step2. Take an analytically well behaved part$(\mathbb{C}-)lamination$$\mathcal{A}_{f}\subset \mathcal{N}_{f}$.

-bundleStep3. Take an.
extends to$\mathbb{R}+$$\mathcal{H}_{f}$of$\mathcal{A}_{f},$the$\mathbb{H}^{3}$

which is a Riemann surface-lamination. Then the action$\hat{f}c\backslash \mathcal{A}_{f}$$f’\backslash \mathcal{H}_{f}$Step4. The extended action$\hat{f}^{Z}r\backslash \mathcal{H}_{f}$is properly discontinuous. Take$\mathcal{M}_{f}:=\mathcal{H}_{f}/\hat{f}$.

Here is the dictionary:FA
prop. distonti.

$f\cap A_{f}$$\mathbb{C}$$\Gamma r^{-}\vee \mathbb{H}^{3}$$\hat{f}\cap \mathcal{H}_{f}$3.1

prop. distonti.

$\mathcal{M}_{f}=\mathcal{H}^{3}/\hat{f}$$M_{\Gamma}=\mathbb{H}^{3}/\Gamma$

Construction of laminations

Interested readers may refer [LM], [L], and [Ka3, \S 2]
Stepl: Natural extension. First we consider all possible backward orbits in our, and, then choose one of its preimage
dynamics. That is, take a point, and so on:choose one of its preimage$z_{0}\in\overline{\mathbb{C}}$$z_{-1}$$z_{-2}$$\overline{\mathbb{C}}\ni z_{0}\underline{f}z_{-1}\underline{f}z_{-2}\underline{f}z_{-3}rightarrow^{f}$...

We gather all of such backward orbits and consider it a subset of
natural extension of is$\overline{\mathbb{C}}\cross\overline{\mathbb{C}}\cross\overline{\mathbb{C}}\cross\cdots$

. The

$f$

$\mathcal{N}_{f}$$:=\{\hat{z}=(z_{0}, z_{-1}, \ldots)\in\overline{\mathbb{C}}^{N}$:$z_{0}\in\overline{\mathbb{C}},$$z_{-n}rightarrow^{f}z_{-n-1}\}$

There is a big merit to consider the inverse limit: We get an invertible dynamics likeas foUows:
Kleninan groups. We define :$f$$\mathcal{N}_{f}arrow \mathcal{N}_{f}$$\hat{f}:(z_{0}, z_{-1}, z_{-2}, \ldots)\mapsto(f(z_{0}), f(z_{-1}),$ $f(z_{-2}),$ $\ldots$$=(f(z_{0}), z_{0}, z_{-1}, \ldots)$$\hat{f}^{-1}$: $(z_{0}, z_{-1}, z_{-2}, \ldots)\mapsto(z_{-1}, z_{-2}, z_{-3}, \ldots)$

)

161

One can easily check that
semiconjugates$f;\mathcal{N}_{f}arrow \mathcal{N}_{f}$$\pi_{f}(\hat{z}):=z_{0}$$\hat{f}$is a homeomorphism. Note that the projection

and :
$f$

$arrow^{f^{\hat}}\mathcal{N}_{f}$$arrow^{f^{\hat}}\mathcal{N}_{f}$$\downarrow\pi_{f}$$\downarrow\pi_{f}$$\downarrow$$arrow^{f}$$arrow^{j}$$\overline{\mathbb{C}}$...

$arrow^{f^{\hat}}$$\overline{\mathbb{C}}$$\downarrow$$arrow^{f}$

Examples.
$f(z)=z^{2}+c$ fixes$\bullet$$\bullet$$\infty$. Thus$:=(\infty, \infty, \infty, \ldots)\in \mathcal{N}_{f}$$\infty\wedge$

satisfies.

$\hat{f}^{\pm 1}(\wedge\infty)=\infty\wedge$Letbe the attracting or parabolic cycle of , withtaking subscript modulo . Thensatisfies$l$$\hat{\alpha}_{k\pm 1}$$\hat{\alpha}_{k}$.

Ifample, for any point$:=(\alpha_{k}, \alpha_{k-1}, \ldots)\in \mathcal{N}_{f}$$z_{0}\not\in\{0, \infty\}$on $J_{fo}=S^{1}$ .

$\hat{f}^{\pm 1}(\alpha_{k})=$thenaccumulates on . For ex, any backward orbit by $f_{0}(z)=z^{2}$ accumulates$\hat{z}=(z_{0}, z_{-1}, \ldots)\in \mathcal{N}_{f}-\{\wedge\infty,\hat{\alpha}_{1}, \ldots,\hat{\alpha}_{l}\}$$\bullet$$f(\alpha_{k})=\alpha_{k+1}$$f$$\{\alpha_{1}, \ldots, \alpha_{l}\}$$J_{f}$$z_{-n}$Remark on H\’enon maps. Though it seems the natural extension is an abstract
object, it naturally appears in 2-dimensional settings. In [HO], Hubbard and Obersteis topologicallyVorth proved that for hyperbolic $f(z)=z^{2}+c$ , the action, whereconjugate to: $(z, w)rightarrow(z^{2}+c-bw, z)$ is a complex H\’enon map.
onwith $|b|\ll 1$ , andis the badkward Julia set of
is an. Note that
Step 2: Affine lamination. Set$f$-invariant set. Moreover, the following properties are known:$f\cap \mathcal{N}_{f}-\{\wedge\infty\}$$F_{b,c}\cap J^{-}$$F_{b,c}$$J^{-}$$\mathbb{C}^{2}$$F_{b,c}$$\mathcal{A}_{f}$1.

$\mathcal{A}_{f}$$:=\mathcal{N}_{f}-\{\wedge\infty,\hat{\alpha}_{1}, \ldots,\hat{\alpha}_{l}\}$$\mathcal{A}_{f}$is a Riemann surface lamination with leaves isomorphic to C.

2. The action$\hat{f}c\backslash \mathcal{A}_{f}$3. Every leaf of$\mathcal{A}_{f}$is a leafwise (complex affine) isomorphism.

is dense in$\mathcal{N}_{f}$.

the affine lamination of .
has an complex affine structure
Step 3:-lamination. Since each leaf ofattached on . In fact, there
isomorphic to , one can consider a hyperbolic space, which is a 3-1amination with leaves isomorphic
ofexists a natural-bundleto. (See Figure 5.) The construction is one of the most technical part. Roughly(up to rotation ofput,
is given as a quotient space of the tangent bundle oftangent vectors). Here we omit the details.
Step 4: Quotient lamination with boundary. An important fact is: The leafvise.
isometry
isomorphic (complex affine) action
extends to a

We call$f$$\mathcal{A}_{f}$$L$$\mathbb{H}^{3}$$\mathbb{C}$$\mathcal{A}_{f}$$L$$\mathbb{H}_{L}^{3}$$\mathbb{R}+$$\mathcal{H}_{f}$$A_{f}$$\mathbb{H}^{3}$$\mathcal{A}_{f}$$\mathcal{H}_{f}$$\hat{f}r\vee A_{f}$$leaf\omega ise$$f\cap \mathcal{H}_{f}$162

Figure 5: A l-dimension down caricature ofand$\mathcal{H}_{f}$$\mathcal{A}_{f}$actsThis corresponds to the Poincar\’e extention. Moreover, the cyclic groupis a Hausdorff space which inherits theproperly discontinuovsly. Thus
. We callthe quotient lamination which correhyperbolic 3-1aminar structure ofsponds toof Kleinian group . As hyperbolic 3-manifolds, the quotient laminations
andmay have “conformal boundary” in the following sense: Set:
decomposition
Fatou-Julia
giveanatural. These sets$\hat{f}^{n}\cap \mathcal{H}_{f}$$\mathcal{M}_{f}$$:=\mathcal{H}_{f}/f$$\mathcal{M}_{f}$$\mathcal{H}_{f}$$\Gamma$$M_{\Gamma}$$\mathcal{F}_{f}$$\mathcal{J}_{f}$$:=(\pi_{f}|_{A_{f}})^{-1}(F_{f})$$;=(\pi_{f}|_{A_{f}})(J_{f})$$\mathcal{A}_{f}=\mathcal{F}_{f}u\mathcal{J}_{f}$$=$$\mathcal{A}_{f}$目$\mathcal{F}_{f}$$\frac{\downarrow}{\mathbb{C}}\pi_{f}$$\mathcal{J}_{f}$$J_{f}\downarrow\pi_{f}$$F_{f}\downarrow\pi_{f}$火$=$properly discontinuously and
It is known that acts on, the conformal boundary oflamination. Set
Here are additional entries to the dictionary:$\hat{f}$$\mathcal{F}_{f}/\hat{f}$$\mathcal{F}_{f}$$\partial \mathcal{M}_{f}:=\mathcal{F}_{f}/f$$\Gamma$(}$\Omega_{\Gamma}$

prop. distonti.

$f$

rv

$\mathcal{F}_{f}$$\mathcal{M}_{f}$is a Riemann surface.

prop. distonti.

$\partial \mathcal{M}=\mathcal{F}_{f}/f$$\partial M_{\Gamma}=\Omega_{\Gamma}/\Gamma$

Actually $f(z)=f_{0}(z)=z^{2}$ (and it perturbation) had been the onlyis divided into two parts, the basin at
well-investigated example(s). In this caseis invariant under
and the basin at zero, D.
infinityis called Sullivan’s solenoidal lamination. There is
the action of , and. Indeed,
has a productcorresponding toofa mirror image$structure\approx S_{0}\cross(0,1)$ that extends to the conformal boundaries. That is, we have
. In [LM] Lyubich and Minsky showed the following:
Example.

$s$$F_{f}$$B_{f}=\overline{\mathbb{C}}-\overline{D}$$\mathcal{B}_{f}$$S_{0}$$\hat{f}$$\overline{S}_{0}$$S_{0}$$:=(\pi_{f}|_{A_{f}})^{-1}(B_{f})$$:=\mathcal{B}_{f}/\hat{f}$$(\pi_{f}|_{A_{f}})^{-1}(D)$$\mathcal{M}_{f}$$\mathcal{M}_{f}\cup\partial \mathcal{M}_{f}\approx S_{0}\cross[0,1]$

Proposition 3.1 If $f_{c}(z)=z^{2}+c$ has an attracting cycle (thus.
largest cardioid of M) then$\mathcal{M}_{f_{C}}\cup\partial \mathcal{M}_{f_{e}}\approx S_{0}\cross[0,1]$$c$

is contained in the

163

$0$

Figure 6: The case of $f=f_{0}$ .

Deformation of. Let us return to a generalwith hyperbolic or parabolic
parameter
M. Setso that $F_{f}=B_{f}uD_{f}$ . Set. Then we have the correspondinganddecomposition. Note that via
:
and
are
conformally identified and so are. This means that if of $f=f_{c}$and$M$moves within , the structure ofis conformally preserved. Hence we also denoteby , and call it the upper end of.
On the other hand, the dynamics $f\cap D_{f}$ is not preserved by the motion of ; eventhe topology ofchanges. In fact, the orbits inare attracted by the attracting or
parabolic cycles and the topology ofreflects the location and multipliers of thesecycles. See Figure 3. The quotient space
upstairs forms a Riemann surface
ation which may have different topology from . We call this end the lower endof.
Now we have some more entries :$f_{0}$$c\in$$\mathcal{B}_{f}$$:=(\pi_{f}|_{A_{f}})^{-1}(B_{f})$$\mathcal{F}_{f}$$=$$D_{f}$$:=(\pi_{f}|_{A_{f}})^{-1}(D_{f})$$\mathcal{B}_{f}u\mathcal{D}_{f}$$B_{f_{0}}arrow B_{f},$$\Phi_{f}$$S_{0}=\mathcal{B}_{f_{0}}/\hat{f}_{0}$$\mathcal{B}_{f}/$$\mathcal{B}_{f}/\hat{f}$$f=f_{c}$$:=K_{\mathring{f}}=\overline{\mathbb{C}}-B_{f}\cup J_{f}$$D_{f}$;$S_{0}$$\mathcal{B}_{f}$$\mathcal{B}_{f_{0}}$$\mathcal{B}_{f}’/\hat{f}$$c$$\mathcal{M}_{f}=\mathcal{H}_{f}/\hat{f}$$c$$D_{f}$$D_{f}$$D_{f}$$S_{f}$$lam\dot{i}$$:=\mathcal{D}_{f}/\hat{f}$$S_{0}$$\mathcal{M}_{f}=\mathcal{H}_{f}/;$FUchsian group$\Gamma$of$S=\mathbb{H}/\Gamma$$M_{\Gamma}\cup(\Omega_{\Gamma}/\Gamma)\approx S\cross[0,1]$

Deformation of4$f_{0}(z)=z$$\mathcal{M}_{f}\cup\partial \mathcal{M}_{f}\approx S_{0}\cross[0,1]$

Deformation of $f_{0}(z)=z^{2}$$\Gamma$A worked out example (Douady’s rabbit)

In this section we mainly deal within Section 2 to describe a typical change ofthe laminations associated with degenerations and bifurcations of hyperbolic quadratic
maps. Let us recall the parameters indicated by the two white dots and the threetriangles in Figure 3. Flrom bottom to top,$f_{1},$$\bullet$$f_{0}(z)=z^{2}$$g,$$f_{2}$, hyperbolic with $f_{0}(0)=0$ .

1In general a family of rational maps caUed Blashcke product corresponds to the family of Fuchsiangroup.

164$\bullet$$f_{1}(z)=z^{2}+c_{1}$, hyperbolic with an attracting fixed point of multiplier$r_{1}e^{2\pi i/3}$

for some $0<r_{1}<1$ .
$\bullet$$\bullet$$9(z)=z^{2}+\sigma$, parabolic with parabolic fixed point of multiplier$f_{2}(z)=z^{2}+c_{2}$$e^{2\pi i/3}$.

, hyperbolic with an attracting cycle of period three whose multi-

plier is $0<r_{2}<1$ .
$\bullet$$f_{3}(z)=z^{2}+c_{rab}$, hyperbolic with $f_{3}^{3}(0)=0$ . (Its Julia set is called “Douady’s

rabbit”.)andOne may expect some topological difference between
but there had
been no investigations before the author’s work [Ka2] and [Ka3]. Here we describe thedifference along the arguments of these papers.
$\mathcal{M}_{fo}$$\mathcal{M}_{f_{S}}$Stability of hyperbolics. First we consider the difference between the laminations(orof andand ). It is known that they have quasi-conformally the same
dynamics near the Julia sets but the whole dynamics are different. The Lyubich-Minsky
laminations have a similar property to the Julia sets. We have:$f_{1}$$f_{0}$$f_{2}$$f_{3}$

andare
Theorem 4.1 ([Ka3]) The dynamics
Indeed,take
isquasiconformal
conjugacy
we canawhich
on theconjugate.
leafivise
hyperboliconthequasi-isometry
andlaminations.
Morelaminations
leafwiseaffine. The same holds forover,
is homeomorphic toand$f_{0}\cap \mathcal{A}_{f_{0}}\cup \mathcal{H}_{f_{0}}$$f_{2}$$\mathcal{M}_{f_{1}}\cup\partial \mathcal{M}_{f_{1}}$$\mathcal{M}_{f_{0}}\cup\partial \mathcal{M}_{f_{0}}$$f_{3}$$f_{1}c\sim \mathcal{A}_{f_{1}}\cup \mathcal{H}_{f_{1}}$.

So it is enough to compare

4.1$f_{1},$

$g$

, and

$f_{2}$.

Degeneration of rabbits

First we describe the difference between the dynanics downstairs. If and tend to
1, and tends to . They are two distinct process of degeneration. It is convenient, andto use extemal rays to describe these process. Since the dynamics of,
we can pull-back the foliation(the basins at infinity) are conformally the same asby the radial rays (which is invariant under the dynamics of ) to each basin aton
is connected, we define the extemd ray of angle
infinity. In general, whenby$r_{1}$$f_{2}$$f_{1}$$r_{2}$$g$$B_{f}1’ B_{9}$$B_{f}2$$B_{fo}$$f_{0}$$B_{f_{0}}$$\theta\in \mathbb{R}/\mathbb{Z}$$K_{f}$$R_{f}(\theta):=\Phi_{f}(\{re^{2\pi i\theta} :r>1\})$.

Then we have $f(R_{f}(\theta))=R_{f}(2\theta)$ .

.

It is known that the parabolic fixed point of is the landing point of
From toraysof angles 1/7, 2/7 and 4/7. For , the landing points of the external
the externalrays of angle angles 1/7, 2/7 and 4/7 $are$ all distinct and form a repemg periodic cycle.
and the repelling cycle degeneratetends to g) the attracting fixed point ofAsinto one parabolic fixed point . See Figure 4, from the left to the middle.
$f_{1}$$g$$\beta$$f_{1}$$f_{1}$$f_{1}$$\beta$$g$

165

In this case, we can find an -invariant graph that joins the central attracting fixed
point and the repelling cycle. we denote it by(Figure 7).
From
to . In the case of , the external rays of angles 1/7, 2/7 and 4/7 landat the same repelling fixed point. As tends to , the repelling fixed point and threeattracting periodic point degenerate into one parabolic fixed point . See Figure 4,
from the right to the middle.
We can aJso find an -invariant graph that joins the central repelling fixed point
and the attracting cycle. we denote it by (Figure 7).
$f_{1}$$I_{1}$$f_{2}$$f_{2}$$g$$g$$f_{2}$$\beta$$f_{2}$$I_{2}$$f_{1}(A_{I_{1}}^{\gamma}\vee$$f_{2}(\hat{\gamma}_{y_{I_{2}}}$$g_{Q_{\bullet}}\beta$

Figure 7: The invariant arcs

and$I_{1}$$I_{2}$.

Now we set
for $i=1,2$ , and $:=\cup g^{-n}(\{\beta\})$ . Figure 8 shows thefilled Julia sets ofwithdrawn in. As we will justify in Theorem 4.3, we may
to .
dynanics
consider that theof is given by just pinching$I_{f:}$$:=\cup f_{i}^{-n}(I_{i})$$K_{f_{i}}$$I_{g}$$I_{f_{l}}$$g$$I_{f_{l}}$$I_{g}$.

’$/\prime \text{ノ^{}\prime}$$\backslash$$\backslash$

$/’=$
’
/

’.,

$/\prec/’$ メ’”$d\prime’’$$”\prime d’$$/^{J’}$’$//’\nu\prime’$$\prime’/’$?

/”Figure 8: The external rays of angles {1/7, 2/7, 1/7} and {1/28, 23/28, 25/28} are also
drawn in.
(the basin at infinity, “outside” the Julia sets)
Tessellation. The dynamics onand their relation $f(R_{f}(\theta))=R_{f}(2\theta)$ . Onrays
is perfectly organized by the external(inside’ the Julia sets) are or perfectly organized by
the other hand, dynamics ontiles.
$B_{f}$$D_{f}$

Theorem 4.2 ([Ka2]) For $i=1,2,$
tiles of the ffom $T_{o}(\theta,m, *)$ with $0=f_{i}$ or $g,$
the following.

$D_{fi}-I_{f_{1}}$and $D_{g}=D_{9}-I_{g}$ are tessellated by$\theta\in \mathbb{Q}/\mathbb{Z},$$m\in \mathbb{Z},$$*=+or-satishing$166$\bullet$$\bullet$$\bullet$$\bullet$

Each tile has a topological disk as its interior.

maps$f_{i}$$g$maps

to $T_{f_{i}}(2\theta, m+1, \pm)$ homeomorphically.

$T_{f_{i}}(\theta, m, \pm)$

to $T_{g}(2\theta, m+1, \pm)$ homeomorphically.

$T_{g}(\theta, m, \pm)$$T_{f}i(\theta, m, *)$and$T_{g}(\theta’, m’, *’)$

$T_{f*}(\theta’, m’, *’)$.

share their boundary points iff so do$T_{g}(\theta, m, *)$andand . It is difficult to draw a nioe picture
Figure 9 shows the tessellations forof the tessellation for because moire appears. But one can imagine it by pinching .
$f_{2}$$f_{1}$$I_{f:}$$g$$2\Pi$’$y$$t/\gamma$$f/\gamma$/$4\Pi$$\theta 7$

Figure 9: The tessellations for$f_{1}$

and$f_{2}$.

Actually the parameters of tiles are closely related to the angles of external rays
but here we omit the details. Interested readers may refer [Ka2].
Pinching semiconjugacies. By using tessellations, we have:$\theta$

Theorem 4.3 ([Ka2]) For $i=1,2$ , there:
such that:$e$tists a continuous and $su7jective$ map$\overline{\mathbb{C}}arrow\overline{\mathbb{C}}$$h_{i}$

is a semiconjugacy. That is, $f_{i}\circ h_{i}=h_{i}og$ .

1.

$h_{i}$

2.

$h_{i}$

sends$I_{f}l$S.

$h_{i}$

sends$T_{f_{2}}(\theta, m, \pm)$4.

$h_{i}|_{B_{f_{1}}}=\Phi_{g}0\Phi_{f:}^{-1}$5.

$h_{i}$

6.

$h_{i}arrow id$

sends$J_{f}l$

as

onto

onto

$I_{g}$. In paritcular, if cardto$T_{g}(\theta,m, \pm)$$h_{i}^{-1}(y)\geq 2$$f_{1}arrow g$$y\in I_{g}$.

homeomorphically.

and it conformally conjugates$J_{g}$

then$f_{i}\cap B_{f_{l}}$to $g\cap B_{g}$ .

. Moreover, it is a conjugacy when $i=2$ .

.

So we can describe the dynamics of downstairs by that of by means of thesemiconjugacies. We can lift the semiconjugacies to the objects (laminations) upstairs.
$g$$f_{i}$

167

4.2

Degeneration of laminationsFirst we roughly describe the degeneration of the affine laminations. By lifting thesemiconjugacy $\hat{h}_{i}(i=1,2)$ to the affine laminations, we can preciselyanddescribe the difference. The lift is explicitly given by:$\mathcal{A}_{f:}$$\hat{h}_{i}$:$(z_{0}, z_{-1}, \ldots)rightarrow(h_{i}(z_{0}), h_{i}(z_{-1}),$$\mathcal{A}_{9}$$\ldots$).

It is convenient to consider the principal and $non- p\dot{n}ncipal$ part of the affine laminations. The principd part
and
and
of
are the sets of backward orbits
accumulating on -invariant arc and the parabolic fixed point respectively. The
rest is called the non-principal part.
$\Lambda_{f_{i}}$$\Lambda_{9}$$I_{i}$$f_{i}$$A_{f}$$A_{9}$$\beta$AMne laminations: Non-principal part. The lifted semiconjugacy $\hat{h}_{i}(i=1,2)$.

. Sincesends the non-principal part ofjustto the non-principal part ofpinchesto , its natural lifton the non-principal parts pinches the backward
orbits withinto those within . As in Figure 10, such a pinching does not change
the structure of leaves so the non-principal part have no significant difference.
$\mathcal{A}_{f}$$I_{f_{i}}$$h_{i}$$\mathcal{A}_{g}$$\hat{h}_{i}$$I_{g}$$I_{f:}$$I_{g}$

Figure 10: Pinching map in the $non- pr\dot{i}$cipal partAffine laminations: Principal part. In the principal parts we have a differenceof topologies. Figure 11 shows the dynamics near $I_{i}(i=1,2)$ and , and theircorresponding principal parts.
The principal partconsists of the three cyclic leaves corresponding to the repelling periodic points of period three. One can regard the action on these leaves as
hyperbolic affine maps by takiUg suitable uniformizations of leaves.
also consists of the three cyclic leaves corresponding to the
The principal part
repelling directions of the parabolic fixed point. One can regard the action on these$\beta$$\Lambda_{f}1$$\Lambda_{g}$168$\backslash _{\backslash }\subset A_{t}’\supset\nearrow 0_{t^{Q^{\backslash }}}\Im\dagger\eta_{\nearrow}$$\subset^{\theta_{\dagger}^{t}O_{\swarrow}}\backslash _{\backslash \wedge}\swarrow 0_{t^{Q^{\searrow}}}^{\iota^{\backslash }}’\bullet\supset$$C,\swarrow o_{t}^{\backslash }Y_{q\backslash }^{\supset}\backslash \}$$\bullet$attract ng$i$$\bullet$parabol ic$O$repel 1 ng$i$$\Lambda_{f_{2}}$Figure 11: The principal part and corresponding dynantics downstairs.
leaves as parabolic affine maps (translations) by taking suitable uniformizations ofis irregular so we can not findleaves. Recall that the backward orbit
it in the affine lamination.
consists of one invariant leaf corresponding to the repelling
The principal part
fixed point. One can aiso regard the action on this leaf as a hyperbolic affine map bytaking a suitable uniformization.
The thickened curves in the leaves shows the lift of -invariant graph . By theof the pinching map , the thickend curves are pinched to the irregular point
lift
in the affine laminations may say that the curves are just pushed away
(Observers.
to infinity.) This causes the topological or dynamical difference between the principal
parts of the affine laminations. See Section 5 of [Ka3] for more details.
$\hat{\beta}=(\beta, \beta, \beta, \ldots)$$\Lambda_{f_{2}}$$I_{i}$$f_{i}$$\hat{h}_{i}$$h_{i}$$\hat{\beta}$

ation is-laminations. Since the-laminations. Next we consider thean -bundle of the affine laminations, we can extend the pinching map .
As in the affine laminations, the 3-dimensional extension of the non-principal parts
have no significant difference. The difference of the 3-dimensional extension of theprincipal parts are described in a similar way (Figure 12). See Section 6 of [Ka3] formore details. Here we just note that the action of (resp. ) on the 3-dimensional) is loxodromic (resp. parabolic).
(resp.
) of(resp.
extension$\mathbb{H}^{3_{-}}1am\dot{m}$$\mathbb{H}^{3}$$\mathbb{H}^{3}$$\hat{h}_{1}$$\mathbb{R}_{+}$$f_{i}$$\Lambda_{g}^{h}$$\Lambda_{f}^{h}i$and$\mathcal{M}_{9}$$\Lambda_{g}$Finally let us consider the quotient laminations. For, we define their principal parts by the quotient of 3-dimensional

Quotient laminations.
$\mathcal{M}_{f}(i=1,2)$$\Lambda_{f:}$$\hat{g}$169Figure 12: Pinching$\mathbb{H}^{3}$-leaves.

extensions of the principal parts by the dynamics. Then we have solid tori$(i=1,2)$ and a rank one cusp
associated withassociated withas the principal parts (leaves) of the quotient laminations.
Since on the 3-dimensional extensions of the non-principal parts is a semiconjugacy, we have:$\ell_{f}i\subset \mathcal{M}_{f}l$$\hat{f}_{i}\cap\Lambda_{f:}^{h}$$\ell_{g}\subset \mathcal{M}_{g}$$\hat{g}\cap\Lambda_{g}^{h}$$\hat{h}_{i}$Theorem 4.4 (Non-principal part)homeomorphic.

$\mathcal{M}_{f_{1}}-\ell_{f_{1}},$$\mathcal{M}_{g}-\ell_{g}$, and$\mathcal{M}_{f_{2}}-\ell_{f_{2}}$are allFigure 13: Non-principal parts of the quotient laminations

So the main difference may appear in the principal part. Note that a solid torus
and a rank one cusp have topologically the same interior. Moreover, the upper ends ofthe principal leaves come from conformally the same dynamics in the basins at infinity.
Thus the topological difference is given by the lower ends in the principal leaves.
By investigating the structures of the lower ends by lifting tiles downstairs, we candescribe the difference of the lower ends as follows (Figure 14.):

170$\bullet$

The lower end ofis an annulus with the lift of . There is a simple closed
curve that corresponds to the backward orbits remained on .
$I_{f_{1}}$$\ell_{f_{1}}$$I_{1}$$\bullet$$\bullet$As
is pinched to , the lower end changes its topology. The lower end ofhas countably many components with only two cuspidal parts.
$I_{g}$$I_{f_{i}}$$\ell_{g}$The lower end ofalso has countably many components with only one annular
component. There is also the lift of. There is a simple closed curve on theannular component that corresponds to the backward orbits remained on .
$\ell_{f_{1}}$$I_{j_{2}}$$I_{2}$$\bullet$is (combinatorially)
In particular, compared with the lower end of , that ofsimple.
corresponding
1/3-twisted along theclosed curve
Here 1/-twisted”
tomeans that there are three tiles along the curve and the connection of tiles are
twisted by the amount of one tile.
$\ell_{f_{1}}$$\ell_{f_{2}}$$I_{1}$

Figure 14: Principal parts of the quotient laminations

Remarks.

1. The case we dealt with is the case when the parameter of $f=f_{c}$ moves $hom$
to the center of the 1/3-limb of the Mandelbrot set. More generally, when it-twisted withmoves into $p/q$ -limb, the lower end is (combinatorially)
$c$

$0$

$\tilde{p}/q$

mod

$q$

$\tilde{p}p\equiv 1$.

2. By sinilax observations, when the parameter continuously moves from to
via parabolic parameters, the interior of the quotientother components oflamination is topologically preserved. After infinitely many degenerations andbifurcations, the parameter will converges to a parameter that gives so-called
an infinitely renormalizable quadratic map. It seems natural to expect that thequotient lamination of such a map has topologically the same interior as thatof , but its lower end is totally degenerate. Here is an interesting question:
Can one define the “end invariants” to distinguish the quotient laminations of
infinitely renormalizable maps?
$c$$M^{o}$$f_{0}$$0$171

References[DH] A. Douady and J. H. Hubbard. Etude dynamique des polyn\^omes complexes
II. Pub. Math. d’Orsay 84-02, 85-05, 1984/85.

$I$&[HO] J.H. Hubbard and R.W. Oberste-Vorth. H\’enon mappings in the complex $doma\dot{i}$II: projective and inductive limit of polynomials. Real and Complex Dynamical
Systems, B. Branner (ed.) et al., NATO $Adv$ . Sci. Inst. Ser. C. Math. Phys. Sci.,
464, Kluwer Acad. Publ., 1995.
[KL] V.A. Kaimanovich and M. Lyubich. Conformal and harmonic measures on laminations associated with rational maps. $Mem$ . Am. Math. Soc., 820, 2005.
[Kal] T. Kawahira. On the regular leaf space of the cauliflower. Kodai Math. J. 26(2003) 167-178.

[Ka2] T. Kawahira. Tessellation and Lyubich-Minsky laminations associated with quadratic maps I: Pinching semiconjugacies. Preprint, 2006.
(arXiv: math. $DS/0609280$ )[Ka3] T. Kawahira. Tessellation and Lyubich-Minsky laminations associated withquadratic maps II: Topological structures of 3-1aminations. Preprint, 2006.
(arXiv: math. $DS/0609836$ )[L] M. Lyubich. Laminations and holomorphic dynamics. Lecture notes in “New Direction in Dynamical Systems”, Kyoto, 2002.
[LM] M. Lyubich and Y. Minsky. Laminations in holomorphic dynamics. J. Differ.
Geom. 47 (1997), 17-94.
[Mi] J.Milnor. Dynamics in one complex variable (3rd edition). Annals of Math Studies
160, Princeton University Press, 2006.