数理解析研究所講究録 1269 巻 2002 年 74-92

74
NOTE ON THE PAPER “A NEW TREE ASSOCIATED WITHHERMAN RINGS”
MITSUHIRO SHISHIKURAThe folowing paper “A new tree associated with Herman rings” was originaly written
and submitted for the proceedings of aconference on complex dyanmical systems, heldin Cornell University (Ithaca,, USA) in August 1987, organized by J. H. Hubbard.
However the proceedings have never been published, and the paper did not have achance
to appear. On th occasion of conference “Complex dynamics and related fields” held at
Research Institute for Mathematical Sciences, Kyoto University, October 9-12, 2002, theauthor decided to include the paper in the Proceedings “Kokyurok\"u, although the paper
itself was not presented during the meeting. The only modification made was to include
the volume number for the reference [2].
$\mathrm{N}\mathrm{Y}$This paper was an early report on the results which were later published in[2] M. Shishikura, Trees associated with the configuration of Herman rings, Ergod. Th.

&Dynam. Sys. 9(1989), p.543-560.

This paper also contains original results such as alower bound on the degree of rational
maps realizing given trees (\S 5 Theorem 3) and many examples (\S 3). In this respect, thispaper complements the paper [2].
DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, KYOTO UNIVERSITY, KYOTO 606-8502,
JAPAN-mail address: mitsuOkusm.
$E$$\mathrm{k}\mathrm{y}\mathrm{o}\mathrm{t}\mathrm{o}-\mathrm{u}$$\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}$1991 Mathematics Subject Classification. Primary
Key words and phrases. Complex dynamics, Herman rings, tree

$58\mathrm{F}23;\mathrm{S}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{y}30\mathrm{D}05$.

75

AIIEW TREE ASSOCIATED WITH HERMAN RINQS
Mitsuhiro Shlshikura
Department of Mathematics
Kyoto University
Kyoto, 606, Japan

ABSTRACTAcertain kind of tree and piecewise linear map on it are defined, inconnection with the configuration of Herman rings of rational functions.
Their properties are investigated and several examples of the tree are
Asufficient condition for trees to be realized by rational
shown.
The construction of the tree can be generalized
functions is given.
These trees
to rational functions with attractive periodic points.
tree.
are completely different from Douady-Hubbard’$\mathrm{s}$INTRODUCTION
Let$\mathrm{f}:\overline{l}\star$

Riemann sphere.
($\mathrm{A}_{\mathrm{p}-1}$$=$$\mathrm{p}$a 1)$\dot{1}\mathrm{i}$$\overline{l}$be arational function, where

Suppose that. .
$\mathrm{e}$the Julia set of’),$(\mathrm{J}=0, \ldots,\mathrm{p}-1|.

$\mathrm{f}(\mathrm{A}_{\mathrm{J}})=\mathrm{A}_{\mathrm{J}+1}$

exists aconformal mapping$\phi\circ \mathrm{f}^{\mathrm{p}}\mathrm{o}\phi^{-]}(\mathrm{z})$has acycle of Herman rings$\mathrm{A}_{0},\mathrm{A}_{]}$are distinct connected components of$\mathrm{A}_{\mathrm{J}}$$\mathrm{f}$$\mathrm{f}$is the$\overline{l}=\mathrm{g}\cup\{\infty\}$$\phi:\mathrm{A}_{0}+\mathrm{t}\mathrm{z}\mathrm{e}\mathrm{C}$$=\mathrm{e}^{2\pi \mathrm{i}6}.\mathrm{z}$$(0<_{\mathrm{r}_{0}}<], \Theta \mathrm{e}\mathrm{R}-\mathrm{Q})$:$\mathrm{r}_{0^{<}}|\mathrm{z}|<$]}\mathrm{A}_{\mathrm{p}}=\mathrm{A}_{0})$,$\overline{\mathrm{G}}-\mathrm{J}_{\mathrm{f}}$.. .,$(\mathrm{J}_{\mathrm{f}}$, and there

such that.

In this paper, we are concerned with the configuration of Herman
rings.

Aconfiguration means acyclically ordered collection of oriented

The author is supported by JSPS Fellowships for Japanese Junior Scientists
and by Grant-in-Aid for Encouragement of Young Scientists , No. 62790099$
The Ministry of Education, Science and Culture, Japan.
;Secondary
Mathematics Subject Classification. Primary$58\mathrm{F}$$3\mathrm{O}\mathrm{D}\mathrm{O}5$

76
Jordan curves$\gamma_{0}$,$\mathrm{Y}_{]}$,$\ldots$,$\gamma_{\mathrm{p}-1}$(and$\gamma_{\mathrm{p}}=\gamma_{0}$orientation preserving homeomorphisms of

inversion of the orientation of$\gamma_{\mathrm{J}}$)$\overline{\mathrm{G}}$in$\overline{\iota}$, considered up to, and up tp simultaneous. For example,$.\mathrm{i}\mathrm{f}$$\mathrm{p}=2$, there are two

possibilities, which are indicated below.

Configuration AConfiguration
Figure 1If$\mathrm{p}=3$$\mathrm{B}$, there are 6possibilities (see 53), and there are two even apartfrom the cyclic order and the orientations of$\gamma_{\mathrm{J}}$.

Figure 2

The cycle of Herman rings defines aconfiguration as follows:that$\mathrm{r}_{\mathrm{Q}}<\mathrm{r}<1$, and define

one fix an orientation of$\gamma_{\mathrm{J}}$. Then$\{\gamma_{\mathrm{J}}\}$$\gamma_{0}$$\gamma_{\mathrm{O}}=\phi^{-]}\{\mathrm{z}\mathrm{e}\mathrm{G}$, then$\mathrm{f}$:$|\mathrm{z}|=\mathrm{r}1$

and

Fix

$\mathrm{r}$$\gamma_{\mathrm{J}}=\mathrm{f}^{\mathrm{J}}(\gamma_{0})$

so

. If

induces equivariant orientations ofdefines aconfiguration, which is independent of the

77
choice of$\mathrm{r}$and the orientation of$\gamma_{0}$.

The surgery done in $[1,56]$ suggests that not only the Herman rings

themselves but also their preimages play an important role in the dynamics
of$\mathrm{f}$. To investigate the configuration of Herman rings and their pre-images , we introduce in Si atree

it , whose$\mathrm{T}_{\mathrm{f}}$and apiecewise linear map

on$\mathrm{f}_{\sim}$

derivativeff is integer.

The primitive idea of the construction of the tree is the$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}/$Suppose that there are disjoint annuli (doubly connected regions) on$\overline{\mathrm{g}}$.

Then one can define agraph:–vertices are connected components of the complement of the annuli;–edges are the annuli;–two vertices are connected by an edge if and only if the corresponding
components have acommon annulus whose boundary is contained in them.
This graph does not contain any non-trivial loop. (Since

connected !)So it can be considered as atree.

Figure 3$\overline{1}$is simply

78
In the actual construction in 51, we take into consideration all“essential” preimages of Herman rings (so they might be infinite ingeneral) and also some quantitative information such as the moduli of

annuli.

Conversely, it is possible to construct arational function fromsuch atree under certain condition 8(54).

As another application, we givean estimate from below on the degree of rational function realizing atree(55).

For example, the configuration$\mathrm{B}$above cannot be realized by any

Finally in 56, the tree construction isrational function of degree 3.

generalized to arational function with super-attractive fixed points.

The proofs the results are not given in this paper.

For the proofs,see [2] and [3].

1.
Let$\mathrm{C}=$

.

{

’

$\mathrm{f}^{\mathrm{n}}(\mathrm{z})$$|\mathrm{z}$$\mathrm{A}_{\mathrm{J}}$is critical points of{connected components of$=$$\mathrm{B}=3\mathrm{A}_{0}\cup 3\mathrm{A}_{]}\cup\ldots\cup 9\mathrm{A}_{\mathrm{p}-1}$

$A$

$=$

{

$\mathrm{A}\mathrm{e}\mathrm{A}$Note thatsense.

’$|\mathrm{f}^{\mathrm{n}}(\mathrm{A})$$A_{0}$,$A$ \dag er(An annulusintersectB.)$\mathrm{m}$$\mathrm{R}$etc. be as in Introduction.

{connected components of ($A_{0}=$$A$

and$\mathrm{f}$OF,$\mathrm{A}_{0}$$\mathrm{f}$,$\mathrm{n}\geq 0$–closure
A6$\mathrm{f}^{-\mathrm{n}}(\mathrm{A})|$

Define},$(\mathrm{A}_{0}\cap \mathrm{C})$$A_{0}$,)},$\mathrm{n}\approx \mathrm{O}$

},

and

separates$\mathrm{B}$for all$\mathrm{n}$a0}.

consist of annuli, hence the definition of
Aseparates aset

For each annulusA$\mathrm{e}4$$\mathrm{B}$, if both components of

makes$\overline{\mathrm{G}}-\mathrm{A}$, there exists aconformal mp$\mathrm{t}_{2}\mathrm{e}\mathrm{c}|\mathrm{e}-2\pi \mathrm{m}_{\mathrm{A}}<|\mathrm{z}|<1$$\}$$\phi_{\mathrm{A}^{:\mathrm{A}}}+$

$A$

,

79
where

$\mathrm{m}_{\mathrm{A}}>0$. For$\mathrm{d}(\mathrm{x},\mathrm{y})=$$\mathrm{x}$$|\mathrm{t}$$\mathrm{I}$,se$\overline{\mathrm{G}}$$\mathrm{y}\mathrm{e}$, define
separartes$|\phi_{\mathrm{A}}^{-]}(\mathrm{e}^{-2\pi \mathrm{s}})$$(0,\mathrm{m}_{\mathrm{A}})$and$\mathrm{x}$

} ,
$|$

$\mathrm{y}$$\mathrm{A}\mathrm{e}A$

wheredenote$|\cdot|$I$\mathrm{I}\mathrm{R}\mathrm{A}$$\mathrm{d}$( ,y)$\mathrm{x}$$\mathrm{s}$the Lebesgue measure on

and$<\infty$

is apseudO-metric on$\mathrm{d}(\mathrm{x},y)$$\mathrm{d}(\mathrm{x},\mathrm{z})\mathrm{S}$$\overline{\mathrm{C}}$$+\mathrm{d}(\mathrm{y},\mathrm{z})$,, satisfies

if and only if$\mathrm{x}\sim \mathrm{y}$associateddefineIRA 2a$\mathrm{w}\dot{/\mathrm{t}}h$

Herman rings.

$(\mathrm{x},\mathrm{y})=\mathrm{d}(\tilde{\mathrm{x}},\tilde{\mathrm{y}})$$\mathrm{T}_{\mathrm{f}}$$\mathrm{d}(\mathrm{x},\mathrm{x})=0$$\overline{\mathrm{G}}^{\mathrm{X}}\overline{\mathrm{G}}$,$\mathrm{d}(\mathrm{x},\mathrm{y})=\mathrm{d}(y,\mathrm{x})$$\mathrm{d}(\mathrm{x},\mathrm{y})=0$

Let, where.

$\pi:\overline{l}+$$\mathrm{T}_{\mathrm{f}}$$\tilde{\mathrm{x}}\mathrm{e}$is called the tree$\mathrm{T}_{\mathrm{f}}$$\pi^{-1}(\mathrm{x})$be the natural Projection, andand$\pi^{-1}(\mathrm{y})$$\tilde{\mathrm{y}}\mathrm{e}$$i$ $.\mathrm{e}.$.

, connected,$\mathrm{o}ne-$dimensional finite simplicial complex containing no looP), andThe fact that$i.\mathrm{e}$$\mathrm{T}_{\mathrm{f}}$. every arc in’$\mathrm{T}_{\mathrm{f}}$is ais isometric to an interval).

Although there might be an

infinite number of annuli, they do not cause infinite branching.
because each annulus separates the$\mathrm{B}$$\mathrm{T}_{\mathrm{f}}+$$\mathrm{T}_{\mathrm{f}}$This is, which consists of afinite number ofconnected components.

$\mathrm{f}_{\sim}:$$\mathrm{d}(\cdot, \cdot)$

becomes atree is understood as in the constructionof the graph in Introduction (see Figure 3).

Defineand’is atopologically finite tree (linear metric on $it$ ($\mathrm{d}$.
$\mathrm{T}_{\mathrm{f}}=\overline{\mathrm{G}}/\sim$

where. Moreover,is continuous on$\mathrm{d}(\cdot, \cdot)$$i.\mathrm{e}.$R.

by$\mathrm{f}_{\mathrm{s}}(\mathrm{x})=\pi\circ \mathrm{f}(3\pi^{-]}(\mathrm{x}))$

,

80
where

is the boundary of$\partial\pi^{-1}(\mathrm{x})$LEMMA 3$\mathrm{f}_{\sim}$$\pi^{-]}$

(x)

in

$\overline{\mathrm{g}}$.

is $well-defined$ and continuous.

Note that for general$\mathrm{x}\mathrm{e}$$\mathrm{T}_{\mathrm{f}}$,is not necessarily apoint.

$\pi\circ \mathrm{f}(\pi^{-]}(\mathrm{x}))$

2. BASIC PROPERTIES
Write$\mathrm{R}\mathrm{R}\mathrm{m}1$$\mathrm{d}=\mathrm{a}$$\mathrm{T}=\mathrm{T}$$\mathrm{f}’$,$\mathrm{F}=\mathrm{f},$. Then ($\mathrm{T},\mathrm{d}$,F)

has thefollowing properties(a)

is a topologically finite tree with alinear metric.

$(\mathrm{T},\mathrm{d})$(b)$\mathrm{F}:\mathrm{T}+$(c)

There exist a finite subset$\mathrm{v}alued$$\mathrm{T}$is $con$ tinuous.

function

DF on

Sing(T,F)ofand apositive integer$\mathrm{T}$-Slng(T,F)which is constant on each component,$\mathrm{T}$such that:if$\mathrm{x}$andare in the same component of$\mathrm{y}$$\mathrm{d}(\mathrm{F}(\mathrm{x}),\mathrm{F}(\mathrm{y})$(d) There exist arcs

at
poin

$\mathrm{I}_{\mathrm{J}}$$)=\mathrm{D}\mathrm{F}(\mathrm{x})\cdot \mathrm{d}(\mathrm{x},\mathrm{y})$$\mathrm{I}_{\mathrm{J}}(\mathrm{J}=0, \ldots,\mathrm{p}-1|.

(interior of$\mathrm{I}_{\mathrm{J}}$)\mathrm{I}_{\mathrm{p}}=\mathrm{I}_{0})$Sing(T,F),$\mathrm{T}$.

in$\mathrm{T}$thensuch that:are mutually disjoint and contain no branch$ts\iota$$\mathrm{F}(\mathrm{I}_{\mathrm{J}})=\mathrm{I}_{\mathrm{J}+1}$

,

and

(e)
$\mathrm{T}=\mathrm{J}_{1}\mathrm{n}\mathrm{u}_{\mathrm{e}\mathrm{o}^{\mathrm{F}^{-\mathrm{n}_{(\mathrm{I}\mathrm{n}\mathrm{t}}}}}\mathrm{I}_{\mathrm{J}})$(f) Every end point of$\mathrm{F}^{\mathrm{p}}|_{\mathrm{I}_{\mathrm{J}}}=\mathrm{i}\mathrm{d}$.

.
$\mathrm{T}$is an end point of an$\mathrm{I}_{\mathrm{J}}$.

Here
Sing(T,F)$=${end points, branch points of$\cup\pi\{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}$$\mathrm{T}_{\mathrm{f}}\rangle$

points of$\mathrm{f}\rangle$,$\cdot$81$\mathrm{I}_{\mathrm{J}}=\pi(\mathrm{A}_{\mathrm{J}})$, called periodic arcs.
The

From Lemmas in the previous section, (a) and (b) follow.

assertions (d), (e) and (f) follow immediately, from the construction and

Let us see how (c) holds.

the properties of Herman rings.

Let

an arc which contains neither branch point nor the image by
critical point of$\mathrm{f}$, and suppose that$8\mathrm{J}\subset\cup\pi(\mathrm{A})$

be

of a

$\pi$

. Then$\mathrm{J}\subset \mathrm{T}_{\mathrm{f}}$$\mathrm{R}=\pi^{-1}(\mathrm{J})$is$\mathrm{A}\mathrm{e}A$an annulus and$\mathrm{f}:\mathrm{R}+$degree of the covering ismultiplied byfor$\mathrm{x}$,$\mathrm{y}\mathrm{e}$$\mathrm{k}$. HenceDFSo

J.

is acovering map onto an amulus.

$\mathrm{f}(\mathrm{R})$$\mathrm{k}$, then the modulus of annulus insatisfies$\mathrm{f}_{\sim}$$=\mathrm{k}$on$\mathrm{J}$If the$\mathrm{R}$is$\mathrm{d}(\mathrm{f}_{\mathrm{r}}(\mathrm{x}),\mathrm{f}_{\sim}(\mathrm{y}))=\mathrm{k}\cdot \mathrm{d}(\mathrm{x},\mathrm{y})$.

$\supset^{\mathrm{f}}$$\mathrm{f}*$$\approx$$\mathrm{J}$Figure 4

Let us introduce some notations, which are used in the following

sections.
DEFIl[TI01i

abranch atsuch that if
write

Let
$\mathrm{x}$$B_{\mathrm{x}}=$. For any$\mathrm{b}$$\mathrm{y}\mathrm{e}$$\mathrm{F}_{\#}(\mathrm{b})=\mathrm{b}’${components of$\mathrm{b}\mathrm{e}$$B_{\mathrm{x}}$$\mathrm{T}$Each$\mathrm{b}\mathrm{e}$$B_{\mathrm{x}}$, there exists aunique branch

is sufficiently near

. Also define-{xl}.

$\mathrm{D}\mathrm{F}(\mathrm{b})=$$\mathrm{x}$, then$\lim$$\mathrm{D}\mathrm{F}(\mathrm{y})$$\mathrm{b}\ni \mathrm{y}\star \mathrm{x}$$\mathrm{F}(\mathrm{x})\mathrm{e}$.

$\mathrm{b}^{1}$

are called$\mathrm{b}’\mathrm{e}$$B_{\mathrm{F}(\mathrm{x})}$. Then we

82
3.
Here are some emmples of trees which satisfy$(\mathrm{a})_{-}(\mathrm{f})$of Theorem 1.

Theorem 2in 54 assures that there exist rational functions which realize
these trees.

The degree of such rational functions are denoted by $”\deg$Periodic arcs$(\mathrm{J}=0,1, \ldots,\mathrm{p}-1)$tree on which

are indicated by arrowed segments with numerals$\mathrm{I}_{\mathrm{J}}$. The map

one which sends$\mathrm{I}_{\mathrm{J}}$DF

EXAMPLE 1 (p).

$\geq 2$Letof only periodic arcs$\mathrm{f}^{\mathrm{n}_{\mathrm{t}}}$to$\mathrm{F}$$\mathrm{I}_{\mathrm{J}+1}$.

on each tree is supposed to be the simplest. The doubled line denotes the part of thebe apositive integer.

$\mathrm{p}$$\mathrm{I}_{\mathrm{J}}$$\mathrm{J}$of period

realized by arational function of$\mathrm{p}$$\mathrm{T}$1,consists. This is the simplest tree and$\deg \mathrm{f}=3$

corresponding configuration is shown.

The treeIf$\mathrm{p}$. On the right, the$\mathrm{p}=2$, this is Configuration A

in Introduction.

$\mathrm{T}_{\mathrm{f},\mathrm{P}}$Example 1(p)$\mathrm{T}_{\mathrm{Z}}$$l^{\mathrm{n}\mathrm{p}\ltimes}$$\mathrm{f}$$\mathrm{F}$

Example 2

EXAMPLE 2.

The tree$\mathrm{T}_{2}$corresponds to this tree,is (isometric to) an interval.
$\deg \mathrm{f}=4$.

Configuration$\mathrm{B}$83
XAMPLE 3 .1-3.6.

Listed here are all possible configurations of period

elow each configuration, an example of tree which realizes it is shown.

ote that$\deg \mathrm{f}>3$$\mathrm{T}_{3\cdot \mathrm{t}}=\mathrm{T}_{\mathrm{t},3}$$\mathrm{d}*\#=3$$d_{\mathrm{C}}mathrm{f}=\sigma$

except Example 3.1.

$\acute{d‘}$84$\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{A}\mathfrak{M}\mathrm{L}\mathrm{E}$Let

T4 is of period 4and4. The tree$\mathrm{e}=\mathrm{t}\mathrm{h}\mathrm{e}$length of the periodic arc$\mathrm{I}_{\mathrm{J}}$$\deg$

f $=3$, which is independent of

It is possible to determine the lengths of segments

denote by the same lettersonto$\mathrm{b}$have$\mathrm{a}=\mathrm{b}$

c)

onto

$\mathrm{a}+\mathrm{e}+\mathrm{d})$(resp.

$\mathrm{c}$$\mathrm{I}_{]}\mathrm{U}\mathrm{b}\mathrm{U}\mathrm{a}$(resp.

$\mathrm{a}$),$\mathrm{b},\mathrm{c},\mathrm{d}$$\mathrm{d}=\mathrm{e}+\mathrm{b}+\mathrm{a}$(resp. aU$\mathrm{I}_{0}\mathrm{U}\mathrm{d}$). First,DF $=1,$with).

$\mathrm{i}$$\mathrm{F}$$\mathrm{a},\mathrm{b}$,,$\mathrm{b}=2\mathrm{e}$,a(resp. d)$\mathrm{e}.$$\mathrm{F}$maps segment

DF $=2$ , hence$\mathrm{c}=4\mathrm{e}$.

, which we also

maps segment$2\mathrm{b}=\mathrm{c}$. Solving these equations, we get$\mathrm{a}=2\mathrm{e}$$\mathrm{c},\mathrm{d}$$\mathrm{j}$. , isometrically, hence weSimilarly,

with

.

and

$\mathrm{d}=5\mathrm{e}$.

$\mathrm{b}$(resp.

(resp.
$2\mathrm{c}=$85
EXAMPLE 5.

This tree is of period 16 and looks complicated, however it isrealizable with$\deg \mathrm{f}=3$

Given atree$(\mathrm{T},\mathrm{d},\mathrm{F})$!

4. REALIZATION PROBLEM, it is natural to ask
satisfying$(\mathrm{a})_{-}(\mathrm{f})$whether there exists arational function realizing the tree or not.

Undercertain conditions, one can construct such afunction, by means of surgery(cf. [31 and also [1]).

DEFINITION Amodel for–$\mathrm{X}$($\mathrm{T},\mathrm{d}$is afinite subset ofpoints of$\mathrm{F}$anda$-\{\mathrm{p}_{\mathrm{b}}\}=\mathrm{t}$ $\mathrm{p}_{\mathrm{b}}|\mathrm{b}\mathrm{e}B_{\mathrm{x}}$$\mathrm{I}_{\mathrm{J}}$,F)

is asetcontaining$\mathrm{T}$$(\mathrm{X},\{\mathrm{p}_{\mathrm{b}}\},\mathrm{g})$satisfying:

Sing(T,F), end points, branch;, xeXUF(X)$\}$

is aset of points of$(\mathrm{X}\mathrm{U}\mathrm{F}(\mathrm{X}) )\mathrm{x}_{\overline{\mathrm{G}}}$, such86
that$\mathrm{p}_{\mathrm{b}}\mathrm{e}$for$\mathrm{x}\mathrm{e}\mathrm{X}$$\deg_{\mathrm{X}}\mathrm{g}$$-\mathrm{I}\mathrm{f}$,$\mathrm{b}\mathrm{e}B\mathrm{x}$,$\mathrm{g}(\overline{l}_{\mathrm{X}})=\overline{\mathrm{g}}_{\mathrm{F}(\mathrm{x})}$is the local degree of$3\mathrm{I}_{0}=\{\mathrm{x}_{]}$,

and$\mathrm{x}_{2}1$if$\overline{\mathrm{O}}_{\mathrm{X}}=\mathrm{t}\mathrm{x}\mathrm{l}^{\mathrm{x}}\overline{\mathrm{r}}$is an analytic mapping from$-\mathrm{g}$(are distinct and$\mathrm{P}_{\mathrm{b}}$$\mathrm{X}\overline{\aleph}$,to$B\mathrm{x}^{\dot{1}}$$\mathrm{F}(\mathrm{X})\mathrm{x}\overline{\mathrm{p}}$$\mathrm{g}(\mathrm{p}_{\mathrm{b}})=\mathrm{p}_{\mathrm{F}_{*}(\mathrm{b})}$at x)$\mathrm{g}$$\mathrm{b}\mathrm{e}$such thatand$\deg_{\mathrm{p}_{\mathrm{b}}}\mathrm{g}=\mathrm{D}\mathrm{F}(\mathrm{b})$$\dot{1}$is the branch containing$\mathrm{b}_{1}\mathrm{e}B$$\mathrm{I}_{0}$$\mathrm{x}1$$(1=1,2)$, then$\mathrm{p}_{\mathrm{b}_{1}}$$\mathrm{e}$,,respectively$-\ominus$$-\mathrm{L}\mathrm{e}\mathrm{t}$$(6 \mathrm{e} \mathrm{R}-\mathrm{Q})$$\iota_{\mathrm{X}}\mathrm{e}\mathrm{X}$$|$$\mathrm{X}_{\sim}=$contained in$\mathrm{X}$}.

are the centers of Siegel disks of rotation number$\mathrm{p}_{\mathrm{b}_{2}}$;the forward orbit ofis finite and$\mathrm{x}$All stable regions (see Remark below) of$\mathrm{g}|_{\mathrm{X}_{t^{\mathrm{X}}}\overline{l}}$aresimply connected.

For any

REMARK.

$\mathrm{n}$a0,(maximal domains wheredefined for$\mathrm{X},$$\{\mathrm{p}_{\mathrm{b}}\}$,g), llke those for a slngle ratlonal functlon.

such thatthen($\mathrm{T},\mathrm{d}$$\mathrm{X}=\mathrm{X}_{k}$$(\mathrm{T},\mathrm{d},\mathrm{F})$$(\mathrm{T}_{\mathrm{f}},\mathrm{d},\mathrm{f}_{\wedge})$$i\epsilon 0\varpi etr\mathrm{y}$.

satisfies,F)$(\mathrm{a})_{-}(\mathrm{f})$(hence all points ofand has amodel
must beSing(T,F)

is realizable by arational function , I.e. ,there exists a rational function

by an. So stable regions$\mathrm{g}|_{\mathrm{X}_{\sim}^{\mathrm{X}}\overline{l}}$preperiodic).

such that$\mathrm{X},\mathrm{x}\overline{\mathrm{C}}$are equicontinuous) and Siegel disks are$\mathrm{g}^{\mathrm{n}}$THEOREM 2Suppose that(can be defined on$\mathrm{g}^{\mathrm{n}}$$\mathrm{f}$having Herman rings

defined as in 51 for

It is expected that the condition$\mathrm{A}_{\mathrm{J}}$$\mathrm{X}=\mathrm{X}_{\sim}$$\mathrm{A}_{0},\mathrm{A}_{]}$

is conjugate to,$\ldots,\mathrm{A}_{\mathrm{p}-1}$$(\mathrm{T},\mathrm{d},\mathrm{F})$can be eliminated.

Asphere can be constructed, by gluing together cylinders instead ofcomponents of$\mathrm{T}$-IC

and spheres instead of points of

One can construct the rational function
mapping and aconformal structure$\mathrm{f}$X.

See Figure 5.

, appropriately defining a

87$\ovalbox{\tt\small REJECT}$. It is easy to show that all the trees in 53 have models , hence

they are realizable.

The degree of$\mathrm{f}$,$\deg \mathrm{f}$, can be computed from thenumber of critical points of the model.

5. LOWER BOUND ON THE DEGREE OF RATIONAL FUNCTIONS$\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{I}[] \mathrm{I}\mathrm{Z}\Pi \mathrm{G}$THE TREESIn this section, we give alower bound on the degree of rational
functions realizing trees or configurations.

sense, the complexity of the tree.
(f) and

Sing(T,F)

DEFINITIONFor

The degree measures , in a

We suppose that$(\mathrm{T},\mathrm{d},\mathrm{F})$satisfy$(\mathrm{a})_{-}$

contains all the end points and the branch points of$\mathrm{x}\mathrm{e}\mathrm{T}$

,

,

,

$\mathrm{b}’\mathrm{e}$$|$$\mathrm{m}_{2}(\mathrm{x})=\mathrm{b}\mathrm{e}B_{\mathrm{X}}[(\mathrm{D}\mathrm{F}(\mathrm{b})_{-}1)$$\mathrm{f}\mathrm{i}_{\mathrm{p}^{(\mathrm{x})\int}}$$\mathrm{m}_{1}(\mathrm{x})=\max(_{\mathrm{b}\mathrm{e}B_{\mathrm{X}}\cap \mathrm{F}_{*}^{-]}(\mathrm{b}^{\mathrm{t}})}^{[\mathrm{D}\mathrm{F}(\mathrm{b})}$$\mathrm{n}_{\mathrm{C}}(\mathrm{x})\sim=2\mathrm{m}_{]}(\mathrm{x})_{-}2-\mathrm{m}_{2}(\mathrm{x})$,$\mathrm{n}_{\mathrm{C}}(\mathrm{x})=\{\begin{array}{l}\mathrm{n}_{\mathrm{C}}(\mathrm{x})*\mathrm{l}\mathrm{f}\mathrm{n}_{\mathrm{C}}(\mathrm{x})*\approx 00\mathrm{I}\mathrm{f}\mathrm{n}_{\mathrm{C}}(\mathrm{x})*<01\mathrm{i}\mathrm{f}\mathrm{n}_{\mathrm{C}}(\mathrm{x})\sim<0\end{array}$$\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{n}\mathrm{d}1\mathrm{s}\mathrm{l}\mathrm{s}$$\mathrm{o}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}$$\mathrm{T}\mathrm{t}$88I$\mathrm{n}_{\mathrm{C}}(\mathrm{X})=$

for any subset$\mathrm{n}_{\mathrm{C}}(\mathrm{x})$xeX$\deg \mathrm{F}$

Note here that$\mathrm{x}\not\in$$+ \frac{1}{2}\mathrm{n}_{\mathrm{C}}(\mathrm{T})$

of

T.

.

Sing(T,F), hence the summation inis actually finite.

$\mathrm{n}_{\mathrm{C}}(\mathrm{X})$Suppose that

THEOREM 3$\mathrm{T}_{\mathrm{f}}$for$\mathrm{n}_{\mathrm{C}}(\mathrm{x})=0$

$=1$

X

and

$\mathrm{F}=\mathrm{f}_{\sim}$$\mathrm{f}$is a rational function with Herman rings andare constructed as in Sl.
$\deg \mathrm{f}$Then&deg F.

words, they cannot be realized by rational functions of lower degree.

Onthe other hand, there exists atree for which the equality in the above

estimate cannot hold.

STARTING FROM CONFIGURATIONS.

Given aconfiguration (of Herman rings) , it is also possible to derive
alower bound on the degree of rational function realizing it.
it for Configuration

LEMMA 4(I)Suppose$\mathrm{B}$and Example 3.3.

$B_{\mathrm{x}}=\{\mathrm{b}_{]}$,$\mathrm{b}_{2}1$,

We will see

For this purpose, we need$\mathrm{b}_{]}\neq \mathrm{b}_{2}$. Then$\mathrm{n}_{\mathrm{C}}(\mathrm{x})=\{\begin{array}{l}|\mathrm{D}\mathrm{F}(\mathrm{b}_{1})_{-}\mathrm{D}\mathrm{F}(\mathrm{b}_{2})|ifF_{\}(\mathrm{b}_{]})\neq F_{*}(\mathrm{b}_{2})\mathrm{D}F(\mathrm{b}_{]})+\mathrm{D}\mathrm{F}(\mathrm{b}_{2})if\mathrm{F}_{*}(\mathrm{b}_{]})=\mathrm{F}_{t}(\mathrm{b}_{2})\end{array}$

(ii)

Let

Suppose$\mathrm{L}$$3\mathrm{L}$$=\{\mathrm{x}_{1}, \mathrm{x}_{2}\}$is an open arc inand$\mathrm{b}_{1}\mathrm{e}B$$\mathrm{x}\mathrm{i}$$\mathrm{T}$containing no branching points.

be the branch at$\mathrm{x}_{1}$containing$\mathrm{L}$89$(\mathrm{i}=1,2)$. then$\mathrm{n}_{\mathrm{C}}(\mathrm{L})\geq|\mathrm{D}\mathrm{F}(\mathrm{b}_{]})_{-}\mathrm{D}\mathrm{F}(\mathrm{b}_{2})|$Moreover , $if$

is not inJec $tive$ , then$\mathrm{F}|_{\mathrm{L}}$$\mathrm{n}_{\mathrm{C}}(\mathrm{L})\gtrless$

The proof is easy.

$\mathrm{X}$It follows from (f) that
and$\mathrm{I}_{0}$.

$\mathrm{T}$respects the orientation ofis (isometric to) an interval.

$\mathrm{T}$$\mathrm{I}_{\mathrm{J}}$for

corresponding to Configuration

do not intersect, since$\mathrm{I}_{]}$$\mathrm{n}_{\mathrm{C}}(\mathrm{X})$T.

of

Let us consider atree

CONFIGURATION B.

Furthermore,$\mathrm{D}\mathrm{F}(\mathrm{b}_{]})+\mathrm{D}\mathrm{F}(\mathrm{b}_{2})$There are more general estimates on

several kinds of subsets

B.

.

is continuous and$\mathrm{F}$. See the figure below.

Figure 6Letonto the whole
that

and

$\mathrm{A}=\mathrm{T}-\mathrm{I}_{0}\mathrm{U}\mathrm{I}_{]}$$\mathrm{x}_{\mathrm{J}}\mathrm{e}$int$\mathrm{T}$,$\lambda$$\lambda$$= \max$

Lemma 4(ii) to arcs$\mathrm{D}\mathrm{F}(\mathrm{x})|\mathrm{x}\mathrm{e}\mathrm{A}$-Sing$(\mathrm{T},\mathrm{F})$must be greater than one.
$(\mathrm{J}=0,1)$$\mathrm{I}_{\mathrm{J}}${and$\mathrm{L}_{0}=(\mathrm{x}_{0},\mathrm{x}_{2})$TakeA-Sing(T, ),$\mathrm{F}$$\mathrm{x}_{2}\mathrm{e}$

and

a3,

Since
$\mathrm{x}_{0},\mathrm{x}_{]}$,$\mathrm{F}$maps$\mathrm{T}$A$8\mathrm{O}$$\mathrm{x}_{2}\mathrm{e}$$\mathrm{D}\mathrm{F}(\mathrm{x}_{2})=\lambda$$\mathrm{L}_{]}=(\mathrm{x}_{2},\mathrm{x}_{]})$$+\lambda$$\mathrm{n}_{\mathrm{C}}(\mathrm{L}_{\mathrm{J}})\gtrless 1$}.

, we have. Applying

90
since

is not injective at$\mathrm{F}$$\deg \mathrm{F}$$\mathrm{L}_{\mathrm{J}}\cap 3\mathrm{I}_{\mathrm{J}}$&1. Therefore,&4.

$+. \frac{1}{2}(\mathrm{n}_{\mathrm{C}}(\mathrm{L}_{0})+\mathrm{n}_{\mathrm{C}}(\mathrm{L}_{]}))$

Thus by Theorem 3, Configuration$\mathrm{B}$cannot be realized by any rational

function of degree less than 4.

CONFIGURATION$3\cdot 3$. Let$\mathrm{T}$be atree corresponding to Configuration 3.3.

It is similarly proved thatis an interval and$\mathrm{T}$$\mathrm{I}_{\mathrm{J}}$do not intersect.

Figure 7

Define$|\mathrm{I}_{\mathrm{J}}|$

Since

Aand
and

as in the above figure and let$\mathrm{B}${$\lambda=\mathrm{m}\mathrm{x}$$\mathrm{D}\mathrm{F}(\mathrm{x})|\mathrm{x}\mathrm{e}\mathrm{A}$and$\mathrm{F}(\mathrm{A})\supset \mathrm{B}\mathrm{U}\mathrm{I}_{]}$$\lambda \mathrm{a}$-SIng$(\mathrm{T},\mathrm{F})$},$\mathrm{F}(\mathrm{B})\supset \mathrm{A}\cup \mathrm{I}_{0}\cup \mathrm{B}$

&b

and

$+\mathrm{e}$$\mathrm{u}$$= \max$$\mathrm{a}=|\mathrm{A}|${,$\mathrm{b}=|\mathrm{B}|$-Sing$\mathrm{D}\mathrm{F}(\mathrm{x})|x\mathrm{B}$,$\mathrm{e}=$$(\mathrm{T},\mathrm{F})$}.

, we lxve$\mathrm{u}\mathrm{b}\gtrless \mathrm{a}+\mathrm{e}+\mathrm{b}$.

It follows that$\lambda\approx 1$

since
Take

$\mathrm{a}$,$\mathrm{b},\mathrm{e}>0$$\mathrm{T}$$\mathrm{x}_{1}\mathrm{e}$.

(Ifso that,$\mathrm{u}$$\lambda=1$$\mathrm{x}_{\mathrm{J}}\mathrm{e}$

It

and

$>1$

and
$\mathrm{I}_{\mathrm{J}}$$\mathrm{u}$$=2$$\lambda+\mathrm{u}\geq 4$, then$(\mathrm{J}4,1 ,2)$,,$\mathrm{e}$must be zero.)A-Slng(T, ),$\mathrm{F}$$\mathrm{x}_{3}\mathrm{e}$$\mathrm{D}\mathrm{F}(\mathrm{x}_{3})=$

91$\lambda$

ande$\mathrm{x}_{4}$$\mathrm{B}$-Sing(T,F),$\mathrm{D}\mathrm{F}(\mathrm{x}_{4})=\mathrm{u}$$\lambda-1$$\mathrm{n}_{\mathrm{C}}((\mathrm{x}_{2},\mathrm{x}_{3}))\gtrless$,$\lambda+1$$\mathrm{n}_{\mathrm{C}}((\mathrm{x}_{3},\mathrm{x}_{0}))\gtrless$and$\mathrm{u}-]$$\mathrm{n}_{\mathrm{C}}((\mathrm{x}_{0},\mathrm{x}_{4}))\approx$(Note that$\mathrm{F}$. Then by Lemma 4(1i), we haveis not injective at$\mathrm{y}.$$\mathrm{n}_{\mathrm{C}}((\mathrm{x}_{4},\mathrm{x}_{]}))$

)

.a

].

$\mathrm{u}-$Therefore,$+ \frac{1}{2}((\lambda-1)+(\lambda+1)+2(\mathrm{W}-1)\}\gtrless$$\deg \mathrm{F}\geq 1$,a4$\lambda+11$$\cdot$So Configuration 3.3 cannot be realized by arational function of degreelesthan 4$\mathrm{s}$$\cdot$

It can be similarly checked that each$3\cdot 6$ls the lowes

$t$

ndeg

in Example 3.2 or$\mathrm{f}^{\mathrm{N}}$$3\cdot 4-$

degree of rational functions realizing the configuration.

6. rRALIZATION
Changing suitably the choice of

associated$\mathrm{w}\dot{\iota}\mathrm{t}\mathrm{h}(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r})\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{l}\mathrm{v}\mathrm{e}$and$A_{0}$$\mathrm{B}$,

one can define atreebasins or Siegel disks.

consider$\mathrm{f}(\mathrm{z})=\mathrm{c}(^{\frac{\mathrm{z}}{1+\mathrm{z}}\mathrm{z}}?$Then$\mathrm{f}$with$|\mathrm{c}|>67$

has asuperattractive fixed point

local coordlmle atcoordinate,$\mathrm{t}\mathrm{z}\mathrm{e}\mathrm{G}$,0by which$\mathrm{f}$$|\mathrm{r}<|\mathrm{z}|<\mathrm{r}’\}$$(0<\mathrm{r}<\mathrm{r}’)$There exists an analytic

0.

such that if$A_{0}$A6$A$

and the tree$\mathrm{T}$$\mathrm{z}^{3}$$\mathrm{g}:\mathrm{z}+$. In thisof disjoint annuli of the form
$A_{0}$, thendoes not intersect the orbit of critical points.

At,.

ls conjugate to

one can choose acollection

For example,

Let$\mathrm{g}(\mathrm{A})\mathrm{e}$$A_{0}$$\mathrm{B}=\{0,\infty\}$andA

As in Si,are defined.

In this case, the tree$\mathrm{T}$is an infinite interval$[-\infty,+\infty]=\mathrm{R}\cup\{\underline{*}\infty\}$

,

92
where

$\pi(0)=arrow$following:,=$\infty$\pi(\infty)$$\mathrm{f}_{\sim}(\mathrm{x})=\mathrm{a}-3|\mathrm{x}|$difficult to see that

Cantor set and$\mathrm{J}_{\mathrm{f}}$and$\pi(\underline{*}])=0$

on 11$(\mathrm{a}>0)$$\pi(\mathrm{J}_{\mathrm{f}})=\mathrm{t}\mathrm{a}\mathrm{e}\mathrm{R}|${. The induced map,$\mathrm{f}_{\sim}(\underline{*}\infty)=\mathrm{r}$,is the. It is notis bounded} , which is a$\mathrm{F}^{\mathrm{n}}(\mathrm{x})|ae0\rangle$

itself is homeomorphic to$\mathrm{f}$$\pi(\mathrm{J}_{\mathrm{f}})\mathrm{x}\mathrm{s}^{\mathrm{j}}$.

Figure 8[11 M.Shishikura, On the quasiconformal surgery of the rational functions,Ann. Scient.

Ec,

Norm. Sup. (4) 20 (1987),[2] –, Trees associatedErg. X.
$\mathfrak{k}31$$l$Dyn. Sys.

$q$$\mathrm{W}[] \mathrm{h}$$\mathrm{p}.1-29$.

the configuration of Herman rings ,$(\{\uparrow \mathrm{g}\mathrm{q} )$$?\cdot 5+3$.

$- \zeta\int \mathit{0}$,$-$ , Construction of Herman rings from trees in preparation