数理解析研究所講究録
1042 巻 1998 年 11-31

11

On the Locus of Crossed Renormalization
Johannes Riedl and Dierk Schleicher
Technische Universit\"at M\"unchen

Contents
1
Introduction
2
Background on Quadratic Polynomials
2
2.1 The Mandelbrot Set and Julia Sets .
3
2.2 Dynamic Rays, Parameter Rays and Equipotentials. . . . . 5
2.3 Polynomial-like Maps.
5
..
. . .
2.4 Renormalization.
7
3 Crossed Renormalization: The Immediate Case8
. . .
3.1 The Principal Construction
10
3.2 The Boundary of the Renormalization Locus.
to the $p/q$ -Limb12
3.3 A Homeomorphism from14
3.4 Our Construction is Complete
15
3.5 Internal Addresses
18
4 Crossed Renormalization: The General Case20
5 References12$C_{p,q}^{n}$

Abstract

We describe the subsets of the Mandelbrot set for which the dynamics is crossed
renormalizable. There are countably many connected components of the locus ofcrossed renormalization of any given period, and each of them is canonically homeomorphic to a limb of the Mandelbrot set. We discuss similarities and differences
to the well-known theory of simple renormalization.

1

Introduction

Renormalization has played an important role in various branches of science including
dynamics and physics. Renormalization ideas help to understand universality of certain
dynamical features which are observed in a host of different contexts, and why certain
dynamical properties can be observed at many different scales.
Holomorphic dynamics offers a substantial collection of powerful tools and is thus
a context in which renormalization is understood particularly well. A high iterate of12a holomorphic map, investigated at a small scale, can display very similar dynamical
properties as a polynomial on a global scale. When a parameter of such a dynamicalsystem is varied, there is a tendency for Mandelbrot sets to appear in parameter space:object” in parameter spaces of iterated holomorphic
the Mandelbrot set is a
maps. This phenomenon manifests itself in the well-known appearance of countably many
little embedded Mandelbrot sets within the entire Mandelbrot set. Every little Mandelbrot
set is the locus of parameters for which the dynamics is renormalizable of a certain kind.
The renormalization theory for holomorphic maps was pioneered by Douady andHubbard [DH]. Since the mid-1980s, two kinds of renormalization have been known(now called “disjoint” and -type” renormalization; together, they are known as “simple
renormalizations”) which correspond to slightly different kinds of embedded Mandelbrot($‘ \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{S}\mathrm{a}\mathrm{l}$“$\beta$renorm.

alization systematically,
discovered that
sets. When describing-type” or “crossed renormalithere was a third kind of renormalization (now calledzation”). From the work of Douady and Hubbard [DH], it is well known that simple-renormalization is organized in the form of finitely many little embedded Mandelbrot
sets in an interesting combinatorial way. The attempt to describe crossed renormalization
in a similar way has motivated the present paper.
Our main result is the following. Termino.logy and background will be explained in
Section 2.
$\mathrm{M}\mathrm{c}\mathrm{M}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{n}[\mathrm{M}\mathrm{c}\mathrm{M}]$$‘\zeta\alpha$$n$

Theorem 1 (The Locus of Crossed Renormalizations)
For any period , the locus of crossed -renormalization within the Mandelbrot set consistsof countably infinitely many connected subsets of : there is a finite number of connected
sets corresponding to every period of the crossing point of the little Julia sets. Every
connected component is canonically homeomorphic to a limb of the Mandelbrot set.
$n$

$n$

$\mathrm{M}$We will make more precise statements in Sections 3 and 4 where we describe how tofind these embedded limbs: we will describe a special case in detail in Section 3, and wewill show in Section 4 that the discussion can easily be extended to the general case. A
first description of the renormalization locus will be given along with the construction in
Section 3.1; in Section 3.2, we will give a second description by chopping off subsets of theMandelbrot set until only a component of the renormalization locus remains, and a third,
combinatorial, description in terms of internal addresses will be presented in Section 3.5.
ACKNOWLEDGEMENT. The second author would like to thank Shunsuke Morosawa forhaving organized the interesting workshop on holomorphic dynamics in Kyoto in October
1997. Moreover, he is grateful to Shizuo Nakane and Mitsuhiro Shishikura for having
made the trip to Japan possible.

2

Background on Quadratic Polynomials

2.1

The Mandelbrot Set and Julia Sets

In this paper, we will discuss exclusively quadratic polynomials; they can all be conjugatedto the form
for a unique complex parameter . The filled-in Julia set
forthe polynomial is the set of points in the dynamic plane such that the iterates of do
not escape to infinity, i.e. the sequence $P(Cz),$is bounded. The Julia set is$P_{c}:z\vdasharrow z^{2}+C$$P_{\mathrm{c}}$

$c$

$z$

$K_{c}$$z$$z,$$P_{c}2(Z),$$J_{\mathrm{r}_{\nu}}$$\ldots$

13
is full (which means
the boundary of the filled-in Julia set. Both sets are compact andthat the complement is connected). The dynamics of a rational map is determined to alarge extent by the critical points of the map (those points where the derivative vanishes)
and their forward orbits. One example of this observation is that the filled-in Julia setof a polynomial is connected if and only if it contains all the critical points in of thepolynomial, i.e. if all the critical orbits in are bounded.
is . Therefore, a
For our polynomials $P_{c}(z)=z^{2}+c$ , the only critical point inunder iteration.
is connected iff does not escape tofilled-in Julia set
is the set of all parameters such that the filled-in Julia set
The Mandelbrot set(or equivalently the Julia set ) is connected, so it is often referred to as the locus
of connected quadratic Julia sets or simply as the quadratic connectedness locus. By
fundamental work of Douady and Hubbard, it is known to be compact, connected andfull.
For every period $n\geq 1$ , there are open sets in parameter space for which the polyhas an attracting periodic orbit of period . These sets are contained in thenomialMandelbrot set, and their connected components are called hyperbolic components of theMandelbrot set. For every fixed period, their number is finite. Every hyperbolic component is conformally parametrized by the multiplier of the attracting orbit: the multiplier
map supplies a biholomorphic map between the component and the open unit disk, and it
extends as a homeomorphism to the closures. Every hyperbolic component has a uniquecenter and a unique root: these are the points where the multiplier map takes values, respectively.
$K_{c}$$\mathbb{C}$$\mathbb{C}$$\mathbb{C}$$0$$K_{\mathrm{c}}$$0$$\infty$$\mathrm{M}$$c$$J_{c}$$K_{c}$$n$$P_{c}$$0$$\mathrm{a}\mathrm{n}\mathrm{d}+1$

2.2

Dynamic Rays, Parameter Rays and Equipotentials

Let us consider some parameter $c\in$ M. The dynamics outside the filled-in Julia set
can conveniently be described by dynamic rays (also known as external rays) and equipois full, there is atentials, which are a dynamic variant of polar coordinates. Since
fixing ; it is unique up to rotation andconformal isomorphismin fact, since our polynomials are
positive,is realcan be fixed so that
normalized so that their leading coefficient is 1, the limit will be equal to 1. The map: we haveinto the dynamics ofconjugates the dynamics in$K_{c}$$\varphi_{c}:\overline{\mathbb{C}}-K_{c}arrow\overline{\mathbb{C}}-\overline{\mathrm{D}}$$\infty$$\lim_{zarrow\infty}\varphi_{c}(z)/z$$z\vdash+z^{2}$$\overline{\mathbb{C}}-K_{C}$$\overline{\mathbb{C}}-\overline{\mathrm{D}}$$\varphi_{c}$$(\varphi_{c}(Z))2=\varphi(z2+c)$.

: $0<r<\infty\}$ ) is called( {er.
, the set
For everyat angle . External angles are counted in full turns. By thethe dynamic ray of
conjugation property we have$1$$\theta\in \mathrm{S}^{1}=\mathbb{R}/\mathbb{Z}$$P_{c}$$e^{2\pi i\theta}$$R_{\theta}:=\varphi_{c}^{-}$$\theta$..

$P_{c}(R_{\theta})=R_{2\theta}$.

In our normalization, a dynamic ray is periodic if its angle is rational with odd denominator when written in lowest terms; it is strictly preperiodic if the angle is rational witheven denominator; and it has an infinite forward orbit if the angle is irrational.
is called the equipotential
For any $r\in(0, \infty)$ , the setof at potential . The conjugation property yields$E_{r}:=\varphi_{c}^{-1}(\{e^{r}\cdot e^{2}\pi i\theta :$P_{\mathrm{c}}$\theta\in \mathrm{S}^{1}\})$$r$$P_{c}(E_{r})=E_{2r}$.

The dynamic rays together with the equipotentials form a coordinate system inin which the dynamics is simply doubling of external angles and potentials. It is often$\mathbb{C}-K_{C}$14
useful, but not always possible, to extend this coordinate system to the Julia set. A
dynamic ray at angle is said to land at a point of the Julia set if$\theta$$z$$\lim_{rarrow 0}\varphi_{c}-1(e^{r.2\pi i\theta}e)=z$.

In general, not every dynamic ray needs to land; its limit set will be a connected subsetof the Julia set. It is well known that every dynamic ray at a rational angle lands at
a periodic or preperiodic point of the Julia set; conversely, every repelling periodic or
preperiodic point is the landing point of some dynamic rays with rational angles, and all
the rays landing at the same point have the same periods and preperiods.
Since the Mandelbrot set is compact, connected and full, we can define external rays
and equipotentials of the Mandelbrot set as well. In order to distinguish these rays fromthe rays in dynamical planes, we call them parameter rays. Again, it is known thatall parameter rays at rational angles land. Parameter rays at periodic angles land inpairs at roots of hyperbolic components such that the periods of both angles and of thecomponent are equal, and every root is the landing point of exactly two such parameter
rays. Parameter rays at preperiodic angles land at parameters for which the critical orbit
is strictly preperiodic; such parameters are known as Misiurewicz points. The number ofparameter rays landing at any given Misiurewicz point is positive and finite.
We have seen above that the boundary of any hyperbolic component $W$ of period
can canonically be parametrized by. Any boundary point at a rational$p/q\neq
0/1$
angle”
is a bifurcation point: at this point, a hyperbolic component of period$qn$ is attached, together with an interesting structure of “decorations”
The bifurcation
point is the root of the $qn$-periodic component, and the pair of periodic parameter rays
landing at this root separates the component of period $qn$ from the component of period
and the origin. The open region which is separated from the origin by this parameter
ray pair is called the $p/q$ -subwake of $W$ . The intersection of this wake withis the p/qsublimb of M. The landing point of the two bounding parameter rays is the root of limb
and wake, but we will not consider it part of limb or wake (this convention differs between
various authors). Subwake and sublimb at internal angle $p/q$ of the unique hyperbolic
component of period 1 are called the $p/q$ -wake and $p/q$ -limb of th. Mandelbrot set; the$n$$\mathrm{S}^{1}=\mathbb{R}/\mathbb{Z}$$‘(\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}$$n$$\mathrm{M}$$\mathrm{e}$$p/q$-limb will be called$\mathrm{M}_{p/q}$.

Now letbe a parameter (not necessarily in M) which is not contained in thereal interval [1/4, ). Then
has exactly two fixed points, one of which is the landing
point of the dynamic ray at angle $\theta=0$ . This fixed point is called the -fixed point of .
The other fixed point is the -fixed point; it may be attracting, indifferent, or repelling,
and it may or may not be the landing point of rational dynamic rays. If it is, the periodof these dynamic rays must be some finite number $q\geq 2$ (because the only ray of period
1 is already taken). The combinatorial rotation number of the dynamics of these raysis then $p/q$ for some integer coprime to . It turns out that the subset of for which
the -fixed point is repelling and the landing point of rays with combinatorial rotation
number $p/q$ is exactly the $p/q$ -wake ofas defined above. Between these wakes, there
is still a Cantor set of external angles for which the number of dynamic rays landing at
is infinite.
$c\in \mathbb{C}$$\infty$$P_{c}$$P_{c}$$\beta$$\alpha$$q$$p$$\mathbb{C}$$q$$\alpha$$q$$\mathrm{M}$$\alpha$15, the -fixed point is either attracting or indifferent (which happens in
For any
the interior respectively on the boundary of the hyperbolic component of period 1), or
it is repelling and is in some $p/q$ -limb of ; in the latter case, the -fixed point must.
disconnect the filled-in Julia set. The -fixed point never disconnects$c\in \mathrm{M}$$\alpha$$\mathrm{M}$$c$$\alpha$$K_{c}$$\beta$Polynomial-like Maps

2.3

A polynomial-like map $f:Uarrow V$ is a proper holomorphic map between two bounded,. Such asuch that
open, connected and simply connected domains$d\geq1$. If this degree is 2, we call it a quadratic-like map.
map has a mapping degree
is a sufficiently large disk
Every polynomial becomes a polynomial-like map whenand $U=p^{-1}(V)$ . But often the dynamics of a high iterate of a polynomial, which itself isa polynomial of large degree, can be understood better by restricting it to an appropriate
subset on which the dynamics is polynomial-like of much smaller degree.
The filled-in Julia set $K(f)$ of a polynomial-like mapping : $Uarrow V$ is the set of all
under iteration of . The Julia set $J(f)$ of is thepoints $z\in U$ which never leaveboundary of $K(f)$ . As for actual polynomials, these sets are connected iff all the criticalpoints of have bounded orbits so they are contained in $K(f)$ .
An important statement is the Straightening Theorem of A. Douady and J. Hubbard[DH, Theorem 1]. For details, as well as for the definition of quasiconformal maps, werefer to this paper and to the references given there.
$f$

$U,$

$\overline{U}\subset V$$V\subset \mathbb{C}$

$V$

$p$

$f$

$U$

$f$

$f$

$f$

Theorem 2 (The Straightening Theorem)
Let : $Uarrow V$ be a polynomial-like map of degree $d\geq 2$ . Then there exists a polynomial$P$of degree such that and $P$ are quasiconformally conjugate in a neighborhood of thefilled-in Julia sets $K(f)$ and $K(P),\dot{i}.e$ . there exists a quasiconformal map which maps
a neighborhood of $K(f)$ to a neighborhood of $K(P)$ such that
$f$

$d$

$f$

$\varphi$$\varphi\circ f\circ\varphi^{-1}=P$.

can be chosen so that its complex dilatation vanishes on $K(f)$ . For
The conjugationis connected.
such a , the polynomial is unique up to affine conjugation$\varphi$$\varphi$$\dot{i}fK(f)$$P$$\square$Douady and Hubbard call a quasiconformal conjugation with vanishing complex dilatation on the filled-in Julia set a hybrid equivalence. For. quadratic polynomials in thenormalization $z^{2}+C$ , every parameter represents its own affine conjugation class, so thestraightening map for quadratic-like maps with connected Julia sets takes well-defined
images in the Mandelbrot set.
$c$

2.4

Renormalization

is called -renormalizable if there are neighborhoods $U,$
A quadratic polynomialof the critical point such that the restriction $P_{c}^{n}:Uarrow V$ is a quadratic-like map withfilled-in Julia
connected filled-in Julia set $K$ . This set $K$ is often referred to as theis$n-1,$set’) of the renormalization. Obviously, $P_{c}^{n}(K)=K$ ; for $j=1,2,$different from $K$ : if $P_{c}^{j}(K)=K$ for some $j<n$ , then the critical orbit would visit
would not be
restricted toat least twice during the first iterations of , andquadratic-like.
$V$

$n$

$P_{c}$($‘ \mathrm{l}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{l}\mathrm{e}$$P_{c}^{j}(K)$$\ldots,$

$U$

$n$

$P_{c}$$P_{c}^{n}$

$U$

16
It is known [, Theorem 7.3] that $K$ can meet any(for $1\leq j\leq n-1$ ) at
most at a single point, which is necessarily periodic and repelling, and the period strictly
divides . Such a point will be a fixed point ofwithin the little Julia set. Since thelittle Julia set is connected, the straightening theorem turns it into the Julia set of apolynomial
and it makes sense to ask whether the corresponding fixed point of$P’$ isthe or fixed point. The set $K$ can never meet its forward images both at andat . Accordingly, we have the following distinction.
$P_{c}^{j}(K)$$\mathrm{M}\mathrm{c}\mathrm{M}$$n$$P_{c}^{n}$$P’\in \mathrm{M}$$\alpha$$\beta$$\alpha$$\beta$$[\mathrm{M}\mathrm{c}\mathrm{M}]$

Definition 3 (Types of Renormalization)
An -renormalization of a quadratic polynomial is said to have$n$disjoint type if the little Julia set is disjoint from all its images under at most $n-1$
iterations;$\beta$-type if the little Julia set meets some of its first $n-1$ forward images only at its$\beta$-fixed$\alpha$-fixedpoint;$\alpha$-type if the little Julia set meets some of its first $n-1$ forward images only at its
point.

The first two types are also known as simple renormalizations, while the last type is known
crossed renormalization. If the little Julia set does meet some of its forward images,
then the renormalization is called immediate the intersection point is a fixed point.
$as$$\dot{i}f$

Examples of simple renormalizations are shown in Figure 4 (disjoint case) and in
Figure 1 ( -case). An example of a crossed renormalization is given in Figure 2. Further
examples are shown and discussed in [, Chapter 7].
It is known from the work of Douady and Hubbard [DH] that the locus of simple-renormalization consists of finitely many connected components, each of which is homeomorphic to the entire Mandelbrot set (to be precise, the renormalization locus is homeomorphic to the entire Mandelbrot set only for disjoint type; for -type, the renormalization
locus is homeomorphic to the Mandelbrot set without its root $c=1/4$ , but the homeomorphism can still be extended to the entire Mandelbrot set). A homeomorphism from sucha connected component tois given by the straightening maps for the little Julia setsarising in the renormalization process. The details of this construction have never been
published, but there is a manuscript of P. Haissinski [Ha]. Every connected component of
the simple -renormalization locus is based at a hyperbolic component ofof period
which is the image of the period 1 component, and every hyperbolic component of period
is contained in a connected component of the -renormalization locus.
Any such homeomorphism fromto a connected component of the simple n-renormalization locus is called a tuning map of period ; a tuning map is a branch of the inverseof the straightening map. A connected component of the simple -renormalization locus
corresponding to disjoint type is based at a primitive hyperbolic component, and by recent work of Lyubich, the tuning map is known to be quasiconformal. In the -case, aconnected component of the simple -renormalization locus is based at a hyperbolic component which bifurcates from a hyperbolic component of some period strictly dividing .
Such hyperbolic components have smooth boundaries and are lacking their characteristic
cusps. In the immediate case, these hyperbolic components are immediate bifurcations
from the main cardioid (the period 1 hyperbolic component) of M.
$\beta$$\mathrm{M}\mathrm{c}\mathrm{M}$$n$$\beta$$\mathrm{M}$$n$$\mathrm{M}$

$n$

$n$

$n$

$\mathrm{M}$

$n$

$n$

$\beta$

$n$

$n$

17

Figure 1: An exampl of a simple renormaliza tion of -type.
$\beta$$e$

Further properties of the simple renormalization loci will be mentioned throughout
the paper in comparison with the crossed renormalization loci.

3

Crossed Renormalization: The Immediate Case

If a quadratic polynomial is crossed -renormalizable, then the little Julia set meets
some of its images at periodic points which separate the little Julia sets. We call thisrenormalization immediate if this periodic point has period 1, i.e. it is a fixed point.
which are relatively prime
Fix an integer $n\geq 2$ and consider two positive integers
consisting ofwith $0<p<q$ . We will be interested in the subset of the $p/nq$ -limb of
type.
parameters which are immediately -renormalizable of crossed
To this end, we will first construct a quadratic-like map for every parameter in the$p/nq$ -wake of(Section 3.1). Such a map is renormalizable if the Julia set of this.
polynomial-like map is connected. We will denote the locus of such polynomials by
to the $p/q$ -limb which turns out to be a
The straightening map defines a map fromcontains every immediately
homeomorphism (Section 3.3). We will then show that$p/nq$-limb (Section 3.4) and that
-renormalizable parameter of crossed type in thecan be obtained from the entire $p/nq$ -limb by cutting off subsets bounded by certain pairsof preperiodic parameter rays landing at Misiurewicz points (Section 3.2). Finally, we will
is immediately -renormalizable of crossedshow how to tell whether a parameter in
The general (non-immediate) case of
3.5).
type in terms of internal addresses (Section
crossed renormalization will be dealt with in Section 4.
$n$$p,$ $q$$\mathrm{M}$$n$$\mathrm{M}$$C_{p,q}^{n}$$C_{p,q}^{n}$$C_{p,q}^{n}$$C_{p,q}^{n}$$n$$C_{p,q}^{n}$

$n$

18
3.1

The Principal Construction

For every parameter in the $p/nq$ -wake of the Mandelbrot set, we will now construct a
quadratic-like map from a restriction of the n-th iterate of the original polynomial to an
appropriate dynamically defined subset. We do not require.
$s_{0}\geq
0$Let
be the potential of the critical value (with $s_{0}=0$ iff). Anyequipotential $s>s_{0}$ bounds a simply connected region which we will call $R.(s)$ . It contains
the filled-in Julia set. From now on, fix a potential $s>2^{n}s_{0}$ .
For all parameters in the $p/nq$ -wake, the -fixed point is the landing point of exactly$nq$ dynamic rays which are permuted transitively by the dynamics of the polynomial.
This permutation has combinatorial rotation number $p/nq$ , i.e. every ray jumps over $p-1$
rays onto its
ray, counting counterclockwise. Similarly, theis the landing
point of equally many rays in a symmetric configuration to the rays landing at , and thepolynomial maps
and its rays onto with its rays. None of these rays contains thecritical point (or the rays could not land), and obviously orcannot be the criticalpoint, either. In particular,.
.
The $nq$ rays landing at , together with the $nq$ rays landing at, cut $R(s)$ into
sectors in a symmetric way (see Figure 2). The sector containing the criticalpoint is itself symmetric and meets both andon its boundary; call its closure .
The closures of the remaining nq–l sectors at will be called,in the same
cyclic order as the dynamics on the bounding rays, such that
contains the critical value.
way,
This
the restriction of towill be mapped onto
for $j=0,1,$ . . . , nq-2.
Finally, we label the closures of the remaining nq-l sectors$Z_{n}q-1$ suchbythat $Z_{j}=-Y_{j}$ . Then the restriction of
to $R(s/2)$ maps tofor $j\leq nq-2$ . Since
they have the same image as theand the global degree ofis two, the restrictionofonto anyoris injective for $j\neq 0$ . Let $Z:=Z_{1}\cup Z_{2}$ U. . .
. Thenthe common image ofand $Z_{nq-1}$ , restricted to $R(s/2)$ , is
; again, the map
is injective on both sectors. Finally,
contains the critical point and maps ontoin a-one fashion.
All theandtogether cover $R(s)$ , with overlaps only at their boundaries. The
restriction oftois a quadratic-like map with range $R(s)$ . We will now identify
a smaller subset of $R(s)$ such that the n-th iterate of w.ill be a quadrat.ic-like map withrange contained in $R(s)$ .
As a first step for our quadratic-like maps, define$c$$c\in \mathrm{M}$$c\in \mathrm{M}$$c$$\alpha$$\mathrm{P}^{\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}}-\alpha$$\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}‘ \mathrm{e}$$\alpha$$-\alpha$$\alpha$$-\alpha$$\alpha$$\alpha\neq-\alpha$$-\alpha$$\alpha$$\mathit{2}nq-1$$-\alpha$$\alpha$$Y_{0}$$\alpha$$\mathrm{Y}_{1},$$\ldots$$Y_{nq-1}$$Y_{1}$$Y_{j}$$R(s/\mathit{2})$$Y_{j+1}$$\mathrm{a}\mathrm{t}-\alpha$$P_{c}$$Y_{j}$$\cup Z_{nq-1}$$Z_{j}$$Y_{j}$$\ldots,$$Y_{j+1}$$Z_{j}$$P_{\mathrm{c}}$$Z_{1},$$Y_{nq-1}$$Y_{0}\cup Z$$Y_{0}$$Y_{1}$$\mathrm{t}\mathrm{w}\mathrm{o}-\mathrm{t}_{0}$$Y_{j}$$Z_{j}$$P_{c}$$R(s/\mathit{2})$$P_{c}$

and$\tilde{U}:=(Y_{0}\cup\bigcup_{j=1}^{-}q1(Yjn\cup Z_{jn}))\cap R(s/2^{n})$$\tilde{V}:=Y_{0}\cup\bigcup_{j=1}^{q}-1Yjn\cup\cup Znq-1j=1j$(see Figure 2). Then
: int
is a proper map between the interiors and has
degree 2. However, there are two problems: int
and int
are disconnected; andand have common boundary points along entire ray segments. The first problem canbe cured by adding a small disk around to(for example a round disk with respect
to linearizing coordinates of ), and adding to two disks around andwhich areappropriately smaller so that they map to the added disk in. The second
under
problem can be cured by thickening the boundaries slightly along the bounding rays, forexample along dynamic rays at slightly different angles. These two problems and theirsolutions are quite standard; for details, see for example Milnor [M2].
$P_{c}^{n}$$(\tilde{U})arrow \mathrm{i}\mathrm{n}\mathrm{t}(\tilde{V})$$(\tilde{U})$$\tilde{U}$$(\tilde{V})$$\tilde{V}$$\tilde{V}$$\alpha$$\tilde{U}$$\alpha$$-\alpha$$\alpha$$\tilde{V}$$P_{c}^{n}$19
We will work with regions and which are the interiors of the enlarged and thickened regions and , respectively. Then $P_{c}^{n}:Uarrow V$ is indeed a quadratic-like map inbecomes -renormalizable whenthe sense of Douady and Hubbard. The polynomial
ever the filled-in Julia set of this quadratic-like map is connected. We will show in thenext section that this renormalization is indeed immediate and of crossed type. It may
not be clear at this point that for a parameter to be -renormalizable it is necessary thatour particular construction y..ields a connected Julia set. We will argue in Section 3.4 that
this is indeed the case.
Of course, there is a considerable amount of freedom in the choice of and in the two
thickening steps. However, all choices will yield hybrid equivalent quadratic-like maps. In
particular, the (filled-in) Julia set is independent of all the choices. This implies that thisJulia set is completely contained in . We will call the (filled-in) Julia set of the quadraticis not renormalizable, thelike map just constructed the “little Julia set”. Even when$p/nq$ -limb for which thetheof
be the subset
little Julia set is well-defined. Let, the critical orbit
because forlittle Julia set is connected. Obviously,
will eventually leave the domain of the polynomial-like map so that the little Julia set is
disconnected.
$V$

$U$

$\tilde{U}$$\tilde{V}$$n$$P_{c}$$n$$s$$\tilde{U}$$P_{c}$$C_{p,q}^{n}$$C_{p,q}^{n}\subset \mathrm{M}$Figure 2: The construction of the quadra $tic$-like maps. The
and is $R(s)-(Y_{1}\cup Y_{3}\cup Y_{5})$ .
$\tilde{V}$$c\not\in \mathrm{M}$$d_{om\mathrm{a}}in\tilde{U}$is shaded,

20
3.2

The Boundary of the Renormalization Locus

The loci of simple renormalization are finitely many homeomorphic copies of the Mandelbrot set within itself. Boundary points of these “little Mandelbrot sets” which disconnect
it from the rest of the Mandelbrot set are certain Misiurewicz points, and the subsetofwhich is disconnected by such a point can be chopped off by a pair of parameterrays landing at this Misiurewicz point. These facts are well known and have analogues forcrossed renormalization. We will describe them in this section, partly by way of examples.
$\mathrm{M}$We want to findwithin the limb
. For any parameter within, andeven within the entire $p/nq$-wake of , we can construct the quadratic-like map as in
Section 3.1. Such a parameter is crossed -renormalizable iff the entire critical orbit
remains within
under.
$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$$\mathrm{M}_{p/nq}$$\mathrm{M}$

$n$

$c$

$\tilde{U}=Y_{0}\cup\bigcup_{j}(Y_{jn}\cup Z_{jn})$$P_{c}^{n}$, the critical orbit will always remain within the filled-in Julia set ofSince
so it can escape from only through the set$c\in \mathrm{M}$$P_{c}$,$\tilde{U}$$Z’:=Z-j=1\dot{q}\cup^{1}z-jn$.

We need only be concerned with the first time the critical orbit enters $Z’$ . This will
obviously happen after $sqn$ steps for some integer $s\geq 1$ .
Let us first consider the case $s=1$ . Under qn–l iterations, the sector
maps
homeomorphically onto $(Y_{0}\cup Z)$ (extended out to an appropriate equipotential. In thissection, we will be rather sloppy about equipotentials and ignore necessary restrictions of
the sectors at equipotentials because that will be irrelevant for our considerations). Thereis a unique point
which maps onto
under
, and it is the landing pointof $qn$ strictly preperiodic dynamic rays (compare Figure 3). These rays cut
into $qn$
sub-sectors. For $j=1,2,$$q-1$ , there is exactly one sub-sector which maps onto
, and one sub-sector will map onto . These sub-sectors are distributed evenly
underaround , and if the critical point is contained in one of them, then the critical orbit willsurvive $qn$ iterations in . However, if the critical value is in one of the remaining qn–q
sub-sectors, then it will leave after $n-1$ iterations.
$Y_{1}$$z_{1}\in Y_{1}$$-\alpha$$P_{c}^{qn-1}$$Y_{1}$$Z_{jn}$$\ldots,$$P_{c}^{n}$$Y_{0}$$q$$z_{1}$$\tilde{U}$$\tilde{U}$Now we transfer this configuration into parameter space. The external angles of the$qn$ dynamic rays bounding these sub-sectors are also the external angles of $qn$ parameterrays of the Mandelbrot set which land at a common Misiurewicz point (compare forexample [Sl, Section 4] and cut the complex parameter plane into $qn$ regions. The
region containing the parameter is bounded by exactly the same external angles as the
sub-sector withincontaining the critical value. Of the $qn$ parameter regions, qn–qof them do not intersect, so that the renormalization locus
is contained in the-star formed by the remaining regions. This is the first step of chopping away subsets
ofin order to approximate, and it describes for which parameters the criticalorbit survives the first $qn$ iterations within .
$)$

$c$

$Y_{1}$$C_{p,q}^{n}$$q$$C_{p,q}^{n}$$q$$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$$\tilde{U}$21, then the critical orbit survives another qn–l (respecor someIfis in
$Y_{1}\cup Y_{2}\cup\ldots\cup Y_{qn-1}$ .
In the next step, the criticalwithintively $(q-j)n-1)$ iterations
or , and it escapes whenever it hits a wrong sector . The $qn$
orbit can again visit
can be transported back for $(q-j)n-1$ iteration steps into
dynamic rays landing. Transporting these rays back another qn-l steps,
or containing
the sector
we obtain $qn$ preperiodic dynamic rays within the sub-sector of containing the criticalvalue. These $qn$ rays cut the sub-sector into $qn$ sub-sub-sectors. If the critical value iscontained in of these, then the critical orbit will survive $qn+(q-j)n$ iteration steps
within ; otherwise, it will not. The index depends of course on which of the sectors
the critical orbit visits first. These $qn$ new preperiodic dynamic rays have again $qn$
counterparts in parameter space which land at a common Misiurewicz point, and theyaround. Of thefurther subdivide the parameter region containing.
this Misiurewicz point, only will intersect
This argument can be repeated: in order for the critical orbit to survive one more turnwithin , there is another collection of $qn$ sub-sub-... -sectors, and only of them may
contain the critical value. We get a countable collection of further necessary conditions
which translate into a countable collection of cuts in parameter space along pairs ofparameter rays at preperiodic angles. Conversely, when a parameter is not cut off by.
such a parameter ray pair, then the critical orbit will remain in forever, andThe first few cuts in the dynamic plane and in parameter space are indicated in Figure 3.
All the Misiurewicz points at which the bounding parameter rays land have the property
that the critical orbit will terminate at the -fixed point after finitely many iterations.
we can conclude the following lemma.
From this$P_{c}^{qn}(0)$$Z_{jn}$$Y_{0}$$Z$$Y_{0}$$Z_{j}$$\mathrm{a}\mathrm{t}-\alpha$$P_{c}^{qn}(0)$$Y_{0}$$Z_{jn}$$Y_{1}$$q$$\tilde{U}$$j$$Z_{j}$$C_{p,q}^{n}$$qn\mathrm{s}\mathrm{u}\mathrm{b}- \mathrm{s}\mathrm{u}\mathrm{b}-\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}_{0}\mathrm{n}\mathrm{S}$$C_{p,q}^{n}$$q$$\tilde{U}$

$q$

$c$

$\tilde{U}$$c\in C_{p,q}^{n}$$\alpha$$\mathrm{d}\mathrm{i}\mathrm{S}\mathrm{c}\mathrm{u}\mathrm{S}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{n}_{7}$are Connected and Almost Compact)
Lemma 4 (The Lociis connected and full, and its union with the root of the limb
Any set$C_{p,q}^{n}$$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$is compact.

. This set is com, so that
PROOF. Let be the root ofpairof parameter rays
by
every
a
set,
cut
pact, connected and full. Starting with thisas described above leaves a compact, connected and full set, and the countable nested
intersection of compact connected and full sets is compact, connected and full. Removing
is still connected and full.
the root again, the remaining set$\mathrm{M}_{p/nq}$$r$$\overline{\mathrm{M}}_{p/nq}=\mathrm{M}_{p/nq}\cup\{r\}$$\square$$C_{p,q}^{n}$

The situation is similar to that for simple renormalizations in the immediate case,say of period : in that case, any connected component of the renormalization locus is alittle Mandelbrot set which is separated from the rest of its limb within the Mandelbrot
set by a countable collection of parameter ray pairs landing at Misiurewicz points, andthese Misiurewicz points again have the property that the critical orbit will terminate at
the -fixed point. The difference is that in the simple case, the renormalization locus
does not extend over such a Misiurewicz point, while it does extend in the crossed casein of the $qn$ directions. Any limb at angle $p/qn$ contains a connected component of(the immediate -renormalization locus) and of the simple $qn$ -renormalization locus.
Every Misiurewicz point bounding the latter is also the landing point of parameter raysis larger and extends further, all the way out to “antenna tips” of, butrestricting
is of course homeomorphic
compare Figure 3. The crossed -renormalization locus, and the locus of simple $qn$ -renormalization corresponds to the tuned copy oftounder this homeomorphism.
of period within$n$$\alpha$$q$$n$$C_{p,q}^{n}$$C_{p,q}^{n}$$C_{p,q}^{n}$$n$$\mathrm{M},\cdot$$C_{p,q}^{n}$$\mathrm{M}$$\mathrm{M}_{p/q}$$q$$\mathrm{M}_{p/nq}$22$\mathrm{a}ndp/q=1/3Figure\mathit{3}:Left.\cdot.\cdot thetherenentiorer\mathrm{m}\mathrm{a}Mli_{Z\mathrm{a}}ti_{on}locuSC^{n}\mathrm{a}ndelbrotSetiSp,qsh_{\mathit{0}}Wnin\mathrm{M}_{p/y2}inlq’ igdispl\mathrm{a}htgrey,\cdot thecIOSSededforn=$-renormaliza tion locus is shown in ark grey, and the simple 6-renormaliza tion
locus is shown in black. Right:
dynamic rays corresponding to arameterrays bounding the renormaliza tion locus.
$2$

$d$

$so\mathrm{m}e$3.3

A Homeomorphism from$p$

$C_{p,q}^{n}$to the $p/q$-Limb

For every parameter in the $p/nq$ -wake of the Mandelbrot set, we have constructed aquadratic-like map and we have defined
to be the subset ofwhere this quadratic-like map has connected Julia set, so that the corresponding polynomials are nrenormalizable. The straightening theorem supplies a well-defined map. In
this section, we will argue that is a homeomorphism onto
, the $p/q$ -limb of M.
Recall that for every, the -fixed point of the polynomialis the landing
point of exactly $nq$ dynamic rays. These rays are permuted transitively by the dynamics
of the polynomial, and the combinatorial rotation number of this cyclic permutation is$p/nq$ . Between any pair of adjacent rays, there is a part of
the Julia set of . Our
region contains of these $nq$ sectors between adjacent rays (plus a small neighborhood
aroundextending into all sectors). The map
permutes these sectors transitively
with combinatorial rotation number $np/nq=p/q$ . The straightening map on
thus
takes images within. The root of that limb cannot be in the image of
becausethe -fixed point is rationally indifferent at the root, while we started with a repelling-fixed point and a topological conjugation can never turn a repelling periodic point into
$C_{p,q}^{n}$$\mathrm{M}_{p/nq}$$\chi:C_{p,q}narrow \mathrm{M}$$\mathrm{M}_{p/q}$$\chi$$c\in \mathrm{M}_{p/nq}$$\alpha$$P_{c}$$P_{c}$$U$$q$$\alpha$$P_{c}^{n}$$\chi$$C_{p,q}^{n}$$\mathrm{M}_{p/q}$$\chi$$\alpha$$\alpha$an indifferent one.

The little Julia set is contained in, and the criticalorbit of the little Julia set first visits the sectors,. Therefore, thelittle Julia set meets the interiors of all the sectors, but of no other. The image of the little Julia set will then be contained in
Therefore, the little Julia set and its image are different. The -fixed pointis their only common point and disconnects both of them. The renormalization is hence
indeed of crossed type, and it is obviously immediate.
Our map has the following important properties:$\tilde{U}\subset \mathrm{Y}_{0}\cup Y_{n}\cup Y_{2n}\cup\ldots\cup Y_{(q-1)n}\cup Z$$Y_{0},$$Y_{0},$$Y_{n},$$Y_{n},$$\mathrm{Y}_{2n},$$\ldots$$\mathrm{Y}_{2n},$$\ldots,$$Y_{i}$$Y_{(q-1)n}$$Y_{(q-1)n}$$Y_{1}\cup \mathrm{Y}_{n+1}\cup Y_{2n+1}\cup\ldots\cup$$Y_{(q-1)1}n+\cdot$$\alpha$$\chi$

23
Proposition 5 (The Straightening Map)
. Moreover, its reonto
is a homeomorphism fromThe straightening map
striction to any hyperbolic component is a biholomorphic map onto another hyperbolic
component and extends homeomorphically to the closures, and the map reduces periodsof hyperbolic components exactly by a factor . In particular, all periods of hyperbolic
are divisible by . Conversely, only hyperbolic components will map
components inonto hyperbolic components.
$\mathrm{M}_{p/q}$$C_{p,q}^{n}$$\chi$$n$$n$$C_{p,q}^{n}$

PROOF. We begin with the second statement. Within hyperbolic components, there
are attracting periodic orbits, the multipliers of which are preserved by hybrid conjugations. Since hyperbolic components are well known to be conformally parametrized by themultipliers of the attracting orbits, and these parametrizations extend homeomorphically
onto the closures ([DH1], [M3] or [S1]), the straightening map inherits the same properties. (In fact, if there are any non-hyperbolic components of the interior of , then thestraightening map will still be holomorphic there.) Conversely, only hyperbolic parameters can map onto hyperbolic parameters. Since any little Julia set can meet its images(other than itself) only at repelling periodic points, it follows that the straightening map
reduces periods of hyperbolic components exactly by a factor of .
It is not hard to check that our construction is “continuous” in the following sense:
for our quadratic-like maps,
andyield nearby domains
nearby parameters inand the quadratic-like maps themselves are close on the intersection of their domains.
Douady and Hubbard [DH] show that under these circumstances the straightening map iscontinuous (in degree two only!). Moreover, they show that it is an open mapping over the$p/q$ -limb. This requires some non-trivial arguments, which are now completely standard:
we have to turn our quadratic-like maps into an analytic family defined in a neighborhoodand define the straightening map in this neighborhood. In order to obtain a wellof
defined straightening map even when the Julia set is disconnected, one has to construct
a tubing. With a chosen tubing, the iniage of the map is well-defined, but in the
disconnected case it depends on the tubing. All these constructions are straightforward.
in our case, so that we indeed have a continuous open mappingby showing that its image is both open
First we show that is surjective ontois well known to be connected and the image is. Since the limb
and closed in
non-empty, this makes surjective.
is obviously open because is an open map. That the image is closed
The image of
be a boundary point of the image. Then there
continuity:letfollows almost fromin the.image converging to , and there are points
is a sequence
within M. If is not the. Let
be a limit point of the sequence
with
with
its root is compact, and
union
of
the
because, thenroot of$\chi(c’)=c$ by continuity. However, if
is the root, then the must converge to the root ofat the roots of the $p/q$-wake: this follows from the proof of local connectivity ofand the $p/nq$ -wake (see Hubbard , Theorem I] or Schleicher [S2]). More precisely, if
has an infinite subsequence on the closure of the hyperbolic component
the sequence
will also have an infinite subsequence, then the image sequenceof period $qn$ in, and this sequence willon the closure of the hyperbolic component of period in
converge to the root of the component. Otherwise, there must be an infinite subsequencein sublimbs of these components, and the diameters of these sublimbs tend to zero as the$\mathrm{M}$

$n$

$V$

$U$

$C_{p,q}^{n}$$C_{p,q}^{n}$$\chi$$\chi:C_{p,q}narrow \mathrm{M}_{p/q}$$\mathrm{M}_{p/q}$$\chi$$\mathrm{M}_{p/q}$$\mathrm{M}_{p/q}$$\chi$$C_{p,q}^{n}$$\chi$$c\in \mathrm{M}_{p/q}$$c_{1},$$C_{1}’,$$c$$c_{2}\ldots$$c’$$\chi(c_{i}’)=\mathrm{q}$$(c_{i}’)$$c’\in C_{p,q}^{n}$$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$$c’$$c_{i}$$\mathrm{M}$$\mathrm{M}_{p/q}$$[\mathrm{H}$$(c_{i}’)$$C_{p,q}^{n}$$(c_{i})$$q$$M_{p/q}$$C_{2}^{;},$$c’$$\ldots\in C_{p,q}^{n}$24
parameter tends to the root (the Yoccoz inequality in [H] makes this precise; a qualitative
version of this same statement can be found in [S2], and both are essential ingredients inthe proof of local connectivity ofat parabolic parameters). This finishes the argumentthatis surjective from, and it extends to a continuous surjective map
tobetween the closures of both sets (which are the same sets with the roots added).
Douady and Hubbard [DH] show that is not only an open mapping, but it is what
they call topological holomorphic over
. This implies thathas a local mapping
degree over every, and it is easy to conclude that it inherits a global mapping. But sincedegree overcontains a single hyperbolic component of period $qn$
which maps biholomorphically onto the unique component of period in, and thisimage component has no further inverse images, the mapping degree is 1. The inverse
map
is continuous as well, and is a homeomorphism.
$\mathrm{M}$$M_{p/q}$$C_{p,q}^{n}$$\chi$$\chi$$M_{p/q}$$\chi$$c\in \mathrm{M}_{p/q}$$\mathrm{M}_{p/q}$$C_{p,q}^{n}$$q$$\chi^{-1}$3.4$M_{p/q}$$\chi$$\square$

Our Construction is Complete

In the last sections, we have identified subsetsofcontaining immediately nrenormalizable polynomials of crossed type. We need to show that
contains every
polynomial with these renormalization properties, so that the sets
and the construction above completely describe the -renormalization locus within. This will bedone in this section.
$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$$C_{p,q}^{n}$$C_{\mathrm{p},q}^{n}$$n$$\mathrm{M}_{p/nq}$

Proposition 6 (Completeness of the Construction)
Any polynomial inwhich is immediately -renormalizable of crossed type is contained in.
$n$$\mathrm{M}_{p/nq}$$C_{p,q}^{n}$

PROOF. Letbe such thatis immediately -renormalizable of crossed type.
Then the restriction of
to appropriate open simply connected domains
defines aquadratic-like map: $Uarrow V$ . Let $K’$ be its filled-in Julia set; it is connected. Since therenormalization is immediate and of crossed type, $K’$ and $P_{c}(K’)$ intersect exactly in ,and separates $K’$ . Let
be the number of connected components of $K’-\{\alpha\}$ .
The -fixed point of is repelling and the landing point of at least 2 periodic dynamic
rays; let be the number of these rays. By [, Theorem 7.11], we have $k\geq qn$ . Theserays, together with the rays landing at, cut the complex plane into
simplyconnected closed sectors,, labeled similarly as above in
Figure 2. Since $K’$ is connected, it contains the critical point, so $K’$ intersects the interiorof . It follows that $K’$ intersects int
for $j=1,2,$$q-1$ . This accounts for atleast connected components of $K’-\{\alpha\}$ , and since was defined as the total numberof connected components of $K’-\{\alpha\}$ , these are all the connected components. If was
greater than $qn$ , then $K’-\{\alpha\}$ would have more than connected components because
permutes the rays landing at and thus the sectors between them transitively. This
is not the case, so $k=qn$ . It follows that$c\in \mathrm{M}_{p/q}$$n$$P_{\mathrm{c}}$$U,$ $V$$P_{c}^{n}$$P_{c}^{n}$$\alpha$$q\geq \mathit{2}$$\alpha$$P_{c}$$\alpha$$k$$\mathrm{M}\mathrm{c}\mathrm{M}$$k$$\mathit{2}k-1$$-\alpha$$Y_{0},$$Y_{1},$$\ldots,$$Y_{0}$$Y_{k-1},$$Z_{1},$$\ldots$$Z_{k-1}$$(Y_{jn})$$q$$\ldots,$

$q$

$k$

$q$

$k$

$P_{c}$$\alpha$$K’ \subset Y0\cup\bigcup_{j}q-1=1Y_{j}\cup\bigcup_{=}^{-1}njq1Z_{j}=\tilde{U}n$(the restriction for thefollows from the symmetry of the map). Since the entire criticalorbit ofis contained in $K’$ , it is contained in the sets
andas specified above.
$Z_{i}$$P_{c}^{n}$$Y_{jn}$$Z_{jn}$25
But this means that the critical orbit never leaves the domain of the quadratic-like map
contains indeed
as claimed. The set
as constructed in Section 3.1, so we have
which are immediately -renormalizable of crossed type.
all parameters in$c\in C_{p,q}^{n}$$C_{p,q}^{n}$$n$$\mathrm{M}_{p/nq}$$\square$does not contain any polynomial which is -renormalizable of simple
The limb
type (because simple -renormalization is organized in the form of embedded Mandelbrot
sets based at hyperbolic components of period , and the $p/nq$ -limb contains only hyperbolic components of periods $nq$ or greater; no such embedded Mandelbrot set can extend
into any the $p/nq$-limb). We will see in Section 4 that crossed -renormalization which
is not immediate is always simply $m$ -renormalizable for some $m>1$ strictly dividing ;is the locus, either. Therefore,
this renormalization type cannot occur within, and this renormalization is immediate and of crossedof -renormalization within
type.
$n$

$\mathrm{M}_{p/nq}$

$n$

$n$

$n$

$n$

$C_{p,q}^{n}$$\mathrm{M}_{p/nq}$

$n$

3.5

$\mathrm{M}_{p/nq}$

Internal Addresses

The notion of internal addresses has been introduced in [LS] in order to efficiently describe
the combinatorial structure of the Mandelbrot set. Formally, an internal address is a(finite or infinite) strictly increasing sequence of integers starting with 1, usually written
as $1arrow n_{2}arrow n_{3}arrow n_{4}\ldots$ with $1<n_{2}<n_{3}<n_{4}\ldots$ . It is associated to a parameteras follows: all parameter rays of the Mandelbrot set at periodic angles land in pairs
with equal periods, and the landing point of such a ray pair is the root of a hyperbolic
component of the same period. This parameter ray pair separates the component fromthe origin. All these ray pairs which separate from the origin are totally ordered. Thenis the lowest
is the lowest period of parameter ray pairs separating from the origin;period of parameter ray pairs separating from the ray pair of period , and so on. (If theparameter is exactly on such a periodic parameter ray pair, the corresponding period isstill accepted.) The internal address is finite iff, after finitely many steps, there no longer
is such a separating periodic parameter ray pair. This happens iff the parameter is on, and the final entry in the internal address is thethe closure of a hyperbolic
period of this component (if the parameter is a bifurcation point between two hyperbolic
components, then their two periods show up as the last two entries in the internal address).
In order to distinguish all the parameters in , every entry in an internal address
has to encode in which sublimb of the corresponding component of period the describedparameter is. This sublimb is described by its internal angle $p/q$ . The denominator isredundant, while can be arbitrary (coprime to ) and separates various combinatorially. These “angled internal addresses” distinequivalent sublimbs of ; forguish combinatorial classes (or “fibers, when taking extra care at hyperbolic components)of the Mandelbrot set completely; compare [S2].
A pleasant property of internal addresses is that they encode the combinatorics ofthe parameters they describe. It is shown in [ , Proposition 6.7] that a parameter issimply -renormalizable if and only if the internal address contains the entry , andevery subsequent entry is divisible by . After -renormalization, an internal address$1arrow n_{2}arrow n_{3}arrow\ldotsarrow n_{j}arrow narrow k_{2}narrow k_{3}narrow\ldots$ turns into $1arrow k_{2}arrow k_{3}arrow\ldots$ .
The renormalization is immediate iff the entry follows directly after the initial 1, so theinternal address has the form $1arrow narrow k_{2}narrow k_{3}narrow\ldots$ with $2\leq k_{2}\leq k_{3}\leq\ldots$ . After
renormalization, the internal address becomes $1arrow k_{2}arrow k_{3}\ldots$$c\in \mathrm{M}$

$c$

$c$

$n_{2}$$n_{3}$$c$$n_{2}$$c$$c$$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\sim \mathrm{n}\mathrm{t}$$n$$\mathrm{M}$

$n$

$q$

$q$

$p$

$\mathrm{M}$$\mathrm{s}\mathrm{e}\mathrm{e}[\mathrm{L}\mathrm{S}]$$\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{S},,$)$\mathrm{L}\mathrm{S}$

$n$

$n$

$n$

$n$

$n$

-

26
There is a corresponding statement for crossed renormalizations, which we state firstfor the immediate case.
Proposition 7 (Crossed Renormalization and Internal Addresses)
Let be a parameter of the Mandelbrot set. Then is immediately crossed n-renormalizable if and only its internal address is of the form
$c$

$c$

$1\backslash arrow k_{1}narrow k_{2}narrow k_{3}narrow\ldots$with $2\leq k_{1}<k_{2}<k_{3}\ldots$ , In this case,with $q=k_{1}$ and someand the internal address ofis $1arrow k_{1}arrow k_{2}arrow k_{3}arrow\ldots$ .
$c\in C_{p,q}^{n}$$p$

coprime to ,$q$$\chi(c)$Within the space available, it is not possible to explain all the details of the combinatorial constructions and of the proof. For background, we refer to [LS] and in particular
to the proof of the related Proposition 6.7. That proof can be modified to hold for the
proposition at hand just as well, but here we will outline a different variant of the proof.

PROOF. Let
be a parameter which is immediately -renormalizable of crossed
type and let $1arrow n_{1}arrow n_{2}arrow\ldots$ be its internal address. Then is in a $p/q$ -limb offorcoprime to and thus
$q=n_{1}$ and some
. Sinceis connected, it connects
to the boundary of the main cardioid of M. Any numberin the internal address
corresponds to a hyperbolic component of period , and the two parameter rays landingat its root disconnect. The hyperbolic component of period
inust then also be in, and by Proposition 5,
is divisible by . We have $n_{1}\geq 2n$ because $1arrow n$ describes
hyperbolic components of period immediately bifurcating from the main cardioid, andthese are simply -renormalizable but not-renormalizable. It is easy to see thatevery entry in the internal address ofcomes from an entry in the internal addressof . Since divides
iods of hyperbolic components by , every entry in the internal
address of is divisible by and the internal address ofis as claimed.
Conversely, assume that some parameter is not crossed -renormalizable. To reacha contradiction, assume that the internal address has the form $1arrow k_{1}narrow k_{2}n\ldots$ with. To any internal address, there is a corresponding kneading sequence (compare[LS]; in the context of real kneading sequences, the associated internal address is thewell-known sequence of cutting times investigated by many people). Such a kneading
sequence is an infinite sequence of symbols and 1 (and, when the internal address isfinite, possibly of symbols ), and it always starts with 1. When all the entries
aredivisible by and greater than , then the only entries in the kneading sequence which
may be different from 1 are at positions $2n,$ $3n,$ $4n\ldots$ (compare, Algorithm 6.2]).
However, from the fact that
, we will deduce that the internal address has anentr.y at a position which is not divisible by .
After an initial string of entries 1 in the kneading sequence, the first or is at
position $n_{1}>1$ . Then is in a-limb offor some coprime to . By assumption,. We can then construct a quadratic-like map as$n_{1}=nq$ for some $q\geq 2$ andin Section 3.1. One can read off a substantial part of the kneading sequence from the
dynamics of the critical orbit: if the k-th forward image of thepoint is containedin, then the k-th entry in the kneading sequence is 1; if the image
point is in. . . $Z_{qn-1}$ , then the k-th entry in the kneading sequence is . Only if
the image point is in , one cannot decide without looking more closely.
$c\in \mathrm{M}$$n$$\mathrm{M}$

$c$

$p$

$c\in C_{p,q}^{n}$$q$$C_{p,q}^{n}$$c$$n_{i}$$n_{i}$$C_{p,q}^{n}$$C_{p,q}^{n}$$n_{i}$$n$$n_{i}$$n$$n$$\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{S}\mathrm{S}.\mathrm{e}\mathrm{d}n$$\chi(c)$$\mathrm{p}\mathrm{e}\dot{\mathrm{r}}$

$c$

$n$

$\chi$$\chi(c)$

$n$

$c$

$n$

$c$

$k_{1}\geq \mathit{2}$$0$$\star$$n$$n_{i}$$n$$[\mathrm{L}\mathrm{S}$$c\not\in C_{p,q}^{n}$

$0$

$n$

$0$

$c$

$p/n_{1}$$\mathrm{M}$$p$$\star$$n_{1}$$c\in \mathrm{M}_{p/nq}$$\mathrm{c}\mathrm{r}\mathrm{i}\ddot{\mathrm{t}}\mathrm{i}_{\mathrm{C}\mathrm{a}}1$$\mathrm{Y}_{1}\cup Y_{2}\cup Y_{3}\ldots Y_{qn-1}$$Z_{1}\cup Z_{2}\cup$$0$$Y_{0}$

27, $Y_{(q-1)n}$ , then the next $n-1$
If any point is in one of the sectors; by symmetry,,forward images on its orbit will still be in one of the sectors$Z_{(q)n}-1$ . In particular, if the critical orbit
the same is true if is in one of$q-1$ ), then we can predict a(for $j=0,1,$
or
falls into one of these sectors
string of $n-1$ entries 1 in the kneading sequence.
$Y_{0},$$z$$Y_{n},$$Y_{2n},$$\ldots$$Y_{1},$$Z_{n},$$z$$Z_{2n},$$Y_{qn-1}$$\ldots,$$Z_{jn}$$Y_{jn}$$\ldots$$\ldots,$

If the critical orbit escapes from$\tilde{U}=(Y_{0}\cup\bigcup_{j=1}^{q-1}(Yjn\cup zjn)\mathrm{I}\mathrm{n}R(_{S}/2n)$with not divisible by : that means,, then it must do so through one of theunder(we can exclude that $P_{c}^{kn}(0)=-\alpha$ because
there is a $k>0$ such that). Now the critical orbit “loses its synchronization” with : writingis also in all the$s=jn-s’$ with $0<s’<n$ , it will spend $s’-1$ steps within some for not divisibleor if $j=q$ ). Unless it is(or inby , until at the s’-th step it will be back inin , we will have a string of $n-1$ symbols 1 as explained above, and in the next stepor . If it lands in , we have an entry in the kneading
the orbit reaches anothersequence at a position which is not divisible by (the position is congruent to modulo
must therefore remain in$n)$ , and this is the desired contradiction.
The orbit under
forever, and when visiting , it must do it in such a way so as to generate an entry
1 in .he kneading sequence. We will argue that this is possible only if the critical orbit, and we had excluded that case above.
escapes through the$Z_{s}$$P_{c}^{n}$

$n$

$s$

$-\alpha$$P_{c}^{kn}(0)\in Z_{s}$$\tilde{U}$$Z_{jn}$$Y_{s}$$Y_{0}$$Y_{jn}$

$n$

$s$

$Z$

$Z$

$Y_{jn}$

$Z$

$Z$

$0$

$s’$

$n$

$P_{c}^{n}$$Y_{0}$$\cup Y_{jn}$$\mathrm{t}$$\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}-\alpha$To do this, we have to look at kneading sequences somewhat differently: assume
that the critical value is the landing point of some dynamic ray at angle . Instead
containing theorof determining the kneading sequence by !ooking at the sectors
critical orbit, we can simply look at the forward orbit of under doubling with respect toan appropriate partition. This partition is formed by the two inverses of under doubling:is in the interval $(\theta/2, (\theta+1)/2)$ , then the k-th entry
and $(\theta+1)/2$ . If
the angles
is on the boundary, this entry is , and the k-th entry
in the kneading sequence is 1; if
is in the remaining interval (containing the angle ). This definition makes sense even
if no dynamic ray lands at the critical value. We just have to associate an external angle
to the parameter , and we take the angle of any parameter ray which accumulates at, we can take any ray which accumulates at the boundary of the; ififconnected component of int(M) containing . If there are several such rays, they will all“essentially” yield the same kneading sequence (except for an occasional symbol , which
can be dealt with). For details, see [LS, Sections 5 and 6].
$\theta$$Z_{j}$$Y_{j}$$\theta$$\theta$$2^{k}\theta$$\theta/2$$2^{k}\theta$$\star$

$0$

$0$

$c$

$c$

$\theta$$c\in\partial \mathrm{M}$$c\in \mathrm{i}\mathrm{n}\mathrm{t}(\mathrm{M})$$c$$\star$In our case, it is quite easy to see that the only external angles which generate only
sequence are those of dynamic rays landing at , which means
symbols 1 in theand terminates at . But wethat the critical orbit escapes” through the point
had excluded that case already, and this is the final contradiction. No critical orbit canthus escape from and still have a kneading sequence or an internal address of the typedescribed above, which finishes the proof of the proposition.
$\alpha$$\mathrm{k}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{i}‘ \mathrm{n}\mathrm{g}\zeta$$-\alpha$$\alpha$$\tilde{U}$$\square$

28
REMARK. The proposition specifies how internal addresses behave under crossed renormalization. We had mentioned above that an internal address is unique only when
extended to an “angled internal address”, specifying the internal angles of sublimbs containing the described parameter. Since the renormalization preserves the parametrizationof hyperbolic components by multipliers and thus by internal angles, it also preserves
the angles in the angled internal address: the angled internal address
turns into.
))
This result allows another approach to showing that the straightening map is a homeomorphism fromto: on a combinatorial level, this follows from internal addresses.
If the Mandelbrot set is locally connected, then its topology is completely described byits combinatorics, and we get an actual homeomorphism. Without assuming local connectivity, the problem is reduced to a local one on every fiber and can easily be settled
using well-known properties of the straightening map.
$1arrow k_{1}n_{(/)}p_{1}q_{1}arrow$$k_{2}n_{(p_{2/}}q2)\cdots$$1arrow k_{1(p_{1/}}q1$$C_{p,q}^{n}$4$arrow k_{2(p_{2/2}}q$$\cdots$$M_{p/q}$

Crossed Renormalization: The General CaseIn the last section, we have described the locus of crossed renormalization for the special

case that the little Julia sets cross at a fixed point (the “immediate” case of crossed
renormalization). The general case is that the little Julia sets cross at a periodic pointof some period $m>1$ . It turns out that the general case can conveniently be reduced tothe immediate case.
Theorem 8 (The General Case of Crossed Renormalization)
Let the polynomial
be crossed -renormalizable so that the crossing point of the little
Julia sets has exact period $m>1$ . Thenis simple $m$ -renormalizable and the corresponding quadratic-like map is immediately $n/m$ -renormalizable of crossed type.
Conversely, the image of any crossed $m$ -renormalizable polynomial under a tuning mapof period is crossed $m$ -renormalizable of period km, and the period of the intersection
point of the little Julia sets is multiplied by as well.
$P_{c}$$n$$P_{c}$$k$$k$

PROOF. Let be a periodic point where the little Julia set crosses one of its forward
images, and let $m>1$ be its period. Obviously, $m$ divides , and $m<n$ because if thefirst return map of was already the first return map of the little Julia set, then the little
Julia set could not cross any of its forward images at .
Since disconnects the little Julia set and at least one of its forward images, it must
be the landing point of at least four dynamic rays. The first return map of must thenpermute these rays transitively (compare [M3]
, Lemma 2.4]), so their period is aproper multiple of $m$ .
the dynamic rays of period
or dividing $m$ which do not land alone cut thecomplex plane into some finite number of pieces. We will consider this situation fromthe point of view of the m-th iterate of . Then this partition is formed by the fixedrays of, and each piece is a “basic region” in the sense of Goldberg and Milnor [GM](they exclude the case that some fixed points ofcoincide; the finitely many paraabolic
parameters where this happens can easily be dealt with at the end, removing puncturesin tuned copies of the Mandelbrot set). The little Julia set and all of its images undersome iterate are each contained within a single basic region: otherwise, the little Julia set
$x$

$n$

$x$

$x$

$x$

$x$

$\mathrm{o}\mathrm{r}.[\mathrm{S}1$$\mathrm{A}1\dot{1}$$m$$P_{c}$$P_{c}^{m}$$P_{c}^{m}$29
would have to extend over a landing point of dynamic rays of period $m$ for , and such alanding point itself has period at most $m$ . At such a point, the little Julia set must meetits m-th forward imag.e, so this point must be . But the rays. landing at have periods
greater than $m$ .
, Lemma 1.6], each basic region contains exactly one fixed point of . SinceBy [, any twoof forward images of the little Julia set are fixed points ofpointscrossing
forward images of the little Julia set are in different basic regions, except those which crossbe the basic region containing the criticalat a point on the forward orbit of . Let
contains
for $j=0,1,2,3\ldots$ because
point and thus the little Julia set. Then
all these are contained in the little Julia set or those of its forward images which cross atorbit are contained within different images of the. All the other points on thelittle Julia set and thus within different basic regions.
Let be the subset of which is mapped onto under . We claim that andso that $P_{c}^{m}:Uarrow V$ is aquadratic-like map
can be thickened slightly to two regionswith connected Julia set.
,
To see this, we first transport back $m$ iterations of along the critical orbit
end,
except at the beginning and. This pull-back will avoid
boundary forming
consists ofpartition
map.
the
two
Since
degreeisaso. And since all, it is forward invariant, which implies
fixed rays of, sowithin some forward image of the little Julia set, it follows that all
havecommonmay
happen
and
.
that
It
never
leave
will
of
the critical orbit
boundary points, but this can be cured by a usual thickening procedure as in Section 3.1.
Call the thickened regions and .
is
We then have a quadratic-like map $P_{c}^{m}:Uarrow V$ with connected Julia set, so$m$ -renormalizable. None of the first $m-1$ forward images of the little Julia set can meetthe interior of , so this renormalization is simple.
By construction, the little Julia set for the crossed -renormalization is contained in(by the same argument as above for the critical orbit). This conand thus also inrenormalizability but reduces its period by $m$ . Therefore, thepreservescrossedstruction
renormalized map is still crossed $m/n$-renormalizable; the crossing point of this renormalization now has period one, so this crossed renormalization is immediate. This proves
the first claim.
The proof of the converse statement is straightforward
口$P_{c}$

$x$

$x$

$f$

$\mathrm{G}\mathrm{M}$$P_{c}^{m}$$\tilde{V}$$x$$\tilde{V}$$P_{c}^{jm}(0)$$\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\dot{\mathrm{l}}$$x$$\tilde{U}$$\tilde{V}$$\tilde{V}$$\tilde{U}$$\tilde{V}$$P_{\mathrm{r}}^{m}$$\cdot$$U,$ $V$$\tilde{V}$$P_{c}(0),$$0\in\tilde{U}$$P_{c}$$\tilde{V}$$P_{c}^{m}(0)\in\tilde{V}$$\ldots,$$\tilde{V}$$P_{c}^{n}:\tilde{U}arrow\tilde{V}$$P_{c}^{jm}(0)\in\tilde{V}$$\tilde{U}\subset\tilde{V}$$P_{c}^{m}$$P_{c}^{(j1)}-m(\mathrm{o})\in\tilde{U}$$\tilde{U}$$\tilde{U}$$\tilde{V}$$P_{c}^{m}$

$U$

$V$

$P_{c}$$\tilde{V}$$n$$\tilde{V}$$\tilde{U}$Any crossed renormalization is thus either immediate, or it is the image of an immediate crossed renormalization under a simple renormalization. Crossed n-renormalization
around a periodic point of period can occur only if $m$ strictly divides , and the corresponding locus is then homeomorphic to countably many homeomorphic copies of limbs
of the Mandelbrot set with denominator $n/m$ . This homeomorphism is a restriction of atuning map of period $m$ (which reduces to the identity in the immediate case $m=1$ ).
All the considerations from Section 3 can now be transferred easily to the generalcase. Any connected component of the crossed -renormalization locus around a periodic
point of period $m$ can be obtained from the Mandelbrot set by chopping off subsets ofbounded by pairs of parameter rays at preperiodic angles. We can also state explicitly
which internal addresses allow crossed renormalizations.
$n$

$m$

$n$

$\mathrm{M}$30

Figure 4: Crossed Renormaliza tion in the general case can be redu $ced$ by a simple
renormaliza tion to an immediate crossed renormaliza tion.

Corollary 9 (Internal Addresses for Crossed Renormalizations)
A parameter of the Mandelbrot set is -renormalizable of crossed type around a periodic
strictly divides and its internal address is of the form
point of period $m$ if and only$n$$\dot{i}fm$$n$$1arrow n_{1}arrow\ldotsarrow n_{j}arrow marrow k_{1}narrow k_{2}narrow k_{3}narrow\ldots$with $1\leq k_{1}<k_{2}<k_{3}\ldots$ . Crossed -renormalization turns this internal address intoand preserves the angles in the angled internal address.
$n$$1arrow k_{1}arrow k_{2}arrow k_{3}arrow\ldots$

5$\square$

References[DH1]

A. Douady, J. Hubbard: Etude dynamique des polyn\^omes complexes. Publications
math\’ematiques d’Orsay #2 (1984) and #4 (1985).

[DH]

A. Douady, J. Hubbard: On the dynamics of polynomial-like maps. Ann. Scient. ENS,
18 (1985), 287-343.

[GM]

L. Goldberg, J. Milnor: Fixed points of polynomial maps. Part II.. fixed point portraits.
Ann. Scient. \‘Ec. Norm. Sup., 4e s\‘erie, t. 26 (1993).

[H]

J. Hubbard: Local connectivity of Julia sets and bifurcation loci: three theorems of J.C. Yoccoz. In: Topological methods in modern mathematics, Publish or Perish, Houston(1993).

[Ha]

P. Haissinski: Modulation dans l’ensemble de Mandelbrot. Manuscript, unpublished.

[LS]

E. Lau, D. Schleicher: Internal addresses in the Mandelbrot set and irreducibility ofpolynomials. Preprint #19, Institute for Mathematical Sciences at Stony Brook (1994).

31
[M1]

[M2]
[M3]

[McM]
[R]
[S1]

[S2]

J. Milnor: Dynamics in one complex variable: introductory lectures. Preprint #5, Institute for Mathematical Sciences at Stony Brook (1991).
J. Milnor: Local connectivity of Julia sets: expository lectures. Preprint #11, Institutefor Mathematical Sciences at Stony Brook (1992).
J. Milnor: Periodic orbits, external rays and the Mandelbrot set: expository lectures.
Preprint (1997).
C. McMullen: Complex dynamics and renormalization. Ann. Math. Studies 135,
Princeton Univ. Press (1994).
J. Riedl: enormierung in der Komplexen Dynamik. In: R. Mayer-Spasche (ed.), Nichtlineare Dynamik, FakTUM13, M\"unchen (1997).
D. Schleicher: Rational external rays of the Mandelbrot set, Preprint #13, Institute forMathematical Sciences at Stony Brook (1997).
$\dot{R}$D. Schleicher: On fibers and local connectivity of Mandelbrot and Multibrot sets,
manuscript in preparation.

Johannes Riedl and Dierk Schleicher

Fakult\"at f\"ur Mathematik
Technische Universit\"at M\"unchen
Barer Stra$e 23
D-80290 M\"unchen
riedl@mathemat ik. tu-muenchen. dedierk@mathemat ik. tu-muenchen. deJanuary 1998