The identity of various exceptional sets in complex dynamics and the Nevanlinna theory and its application to the potential theory (Complex Dynamics)

The

identity

of

various

exceptional

sets

in

complex dynamics and the

Nevanlinna

theory

and its application

to

the

potential

theory

Y\^usuke

_Okuyama

*

Department of

Mathematics,

Faculty

of

Science,

Kanazawa

University, Kanazawa

920-1192

Japan

email;

[email protected]

奥山

裕介

金沢大学理学部数学科

17

February

2004

This article is

an

announcement of the preprint [11]. A rational

map

$f$ is

a

holomorphic endomorphism ofthe Riemann sphereC.

Notation, _{Rat denotes the set of all rational endomorphism ofC.} $\hat{\mathbb{C}}$

isidentified

as

the setofallconstantfunctionsofC.

In the

cases

$f$ is non-invertible, theFatou and Julia strategy for studying the

complex dynamics $(\hat{\mathbb{C}},f)$, which treats forward-images under iterations, is the

separation of $\hat{\mathbb{C}}$

into two completely

invariant

complementary subsets,

one

of which istheFatouset $F(f)$, the region ofnormality of$\{f^{k}:=f^{\mathrm{o}k}\}$, andtheother

$\mathrm{t}\grave{\mathrm{n}}\mathrm{e}$Juliaset$J(f)$. In otherwords,the restricted dynamical systems$(F(f), f)$ and

$(J(f),f)$

are

_{tame and chaoticrespectively. Consequently, thedynamical system}

around$J(f)$has

an

almost covering feature: Thereexists$E(f)\subset\hat{\mathbb{C}}$ such that for

every

neighborhood $U$ of

a

point

of_$J(f)$, the union of the forward-images of $U$

under

iterations

covers

$\hat{\mathbb{C}}-\mathrm{E}(\mathrm{f})$

.

Definition

1

(dynamical exceptional set). $E(f)$

is

called the dynamical

excep-tionalsetof

_f.

’Partially supported by the SumitomoFoundation, andbythe Ministry ofEducation,Science,

Sportsand Culture,$\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}- \mathrm{i}\mathfrak{n}$-Aidfor Young Scientists

From this almost coveringfeature, naturally

arises

the Nevanlinnatheoretical study,whichtreats preimagesunder

iterations.

Definition

2

(value distribution). _For$f$,$g$ $\in$ Rat, the value distribution_{$\mu(f, g)$}

of$f$ for$g$

is

defined by the

mass

distribution

on

the $(\deg f+\deg cf)$-roots ofthe

equation $f=g$.

The spherical

area

measure

and thechordaldistance

on

$\hat{\mathbb{C}}$

are

$o^{-}(w)= \frac{|dw|}{\pi(1+|w|^{2})^{2}}$ aannd _{$[z, w]= \frac{|z-w|}{\sqrt{1+|z|^{2}}\sqrt{1+|w|^{2}}}$} respectively. We note thatthey

are

normalized

as

$\sigma(\hat{\mathbb{C}})=1$and $[0, \infty]=1$.

Definition

3

(dynamicalNevanlinnatheory [13]). For$f$,$g$ $\in$ Rat,thepointwise

proximity

_function

is definedby

$(w(g, f))(z):= \log\frac{1}{[cf(z),f(z)]}$

:

$\hat{\mathbb{C}}arrow[0, \infty]$,

and the

mean

proxim

nit}’

by

$m(g,f):= \int_{\hat{\mathbb{C}}}w(g,f)do^{\sim}\in[0, \infty)$.

Let $F$ be

a

rational

sequence

$\{f_{k}\}_{k=0}^{\infty}\subset$ Rat with increasing degrees $\{d_{k}:=$

$\deg f_{k}\}$. For$g\in$ Rat, the dynamicalNevanlinna and Valironexceptionalities

are

definedby

NE(g;$\mathcal{F}^{\cdot}$)

$:=$ $\lim_{karrow}\inf_{\infty}\frac{m(g,f_{k})}{d_{k}}\in[0, \infty]$,

$\mathrm{V}\mathrm{E}(g;F)$ $:=$ $\lim_{karrow}\sup_{\infty}\frac{m(g,f_{k})}{d_{k}}\in[0, \infty]$

respectively.

From

now

on,

we

considerthe iteration

sequence

$\{f^{k}\}_{k=1}^{\infty}$ of

a

rational

map

_$f$

ofdegree$d\geq 2$

.

Definition

4

(dynamical Nevanlinna and Valiron exceptional sets). The dy-namicalNevanlinna and Valironexceptional sets of$f$in$\hat{\mathbb{C}}$

are

defined by

$E_{N}(f):=$

{

$p$

$\in\hat{\mathbb{C}}$;

NE(p;$\{f^{k}\})>0$},

$E_{\mathcal{V}}$($f\rangle$ $:=\{p\in\hat{\mathbb{C}}$;VECp;$\{f^{k}\}$) $>0\}$

We shall

use

several notions from the

geometric

measure

theory and the

po-tential theory. For thedetails, see,forexample, [3], [10], and [7].

Itis known that$\{(f^{k})\mathrm{A}\sigma/d^{k}\}$

converges

weakly. The limit isalso known

as

the

unique maximal entropy

measure

(see [8] and [9]). Definition 5 (themaximalentropymeasure).

$\mu_{f}:=\lim_{karrow\infty}\frac{(f^{\mathrm{A}’})^{*}\zeta\gamma}{d^{\mathrm{A}}}$.

Definition

6

(accumulation and

convergence

loci). Theaccumulation and

con-vergence loci $0\dot{\mathrm{r}}$

theaveraged value disrributions of

f

in $\hat{\mathbb{C}}$

are

definedby

$A(f):=$

_{

$p\in\hat{\mathbb{C}}$;

a

subsequence of$\{\mu(f^{k},p)/d^{k}\}$

converges

to

p7},

Conv(f) $:= \{p\in\hat{\mathbb{C}};\lim_{\mathrm{A}arrow\infty}\frac{\mu(f^{k},p)}{d^{k}}=\mu_{f}\}$

respectively.

Now

we

stateMainTheorem.

MainTheorem

1

(characterizationsofexceptionalsets). For$f\in$Rat

of

degree

$\geq 2$,

$\hat{\mathbb{C}}-Ev(f)=\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}(/)\subset A(f)=\hat{\mathbb{C}}-E_{N}(f)\subset\hat{\mathbb{C}}-E(f)$

.

Independently, known is the following remarkable theorem which

was

first proved forpolynomials by Brolin [1] and later for rational

maps

by Lyubich [8] and independently byFreire-Lopes-Mane \’e[5]. See also [2], [6], [4] for the other proofs.

Theorem 1 (convergence ofaveragedvalue distributions). For$f\in$ Rat

of

de-$gree\geq 2$,

$\hat{\mathbb{C}}-E(_{-}\eta=\Gamma_{-\mathrm{O}_{-}}\eta \mathrm{v}\iota J)$.

Combiningthem,

we

havethefollowing.

Main Corollary

1

(Allexceptional sets

are

same.). For

_f

$\in$ Rat

of

degree $\geq 2_{r}$

$E_{N}(f)=Ev(f)=\mathrm{E}(\mathrm{f})=\hat{\mathbb{C}}$-Conv(/) $)=\hat{\mathbb{C}}-\mathrm{A}(\mathrm{f})$

.

Remark

1.

In [12], Main Corollary

1

has been already implicitly applied to the Siegel-Cremer

linearizability

problem of rational

maps.

The

important consequence

of Main Corollary is

a

convergence

theorem of thepotentials of the averaged valuedistributions

Definition 7 (sphericalpotential). For

a

regular

measure

$\mu$

on

$\hat{\mathbb{C}}$

,the potentialis defined by

$V_{\mu}:= \int_{\hat{\mathbb{C}}}-\log[\cdot,$_$w1\mu(w)$

:

$\hat{\mathbb{C}}arrow[0, \infty]$.

Remark

2.

In thepotential theory, thepotential isusually defined

as

$-V_{\mu}$, but the

definition will be

more

convenient

in

our

study.

The (axiomatic) potential theory impliesthatwhen regular

measures

$\mu_{k}$

con-verges

to$\mu$, then

$\lim_{\mathrm{A}^{r}arrow}\inf_{\mathrm{m}}V_{\mu k}=V_{\mu}$

quasieverywhere

on

C. For the averaged valuedistributions,

we

obtain thestronger conclusion.

MainTheorem2(convergencetheoremofpotentials). Let

_f

$\in$ Rat be

of

degree

$d\geq 2$

.

If

$p\in\hat{\mathbb{C}}-E(f)$ isnot

afixed

point then

$\lim_{karrow}\inf_{\infty}V_{\mu(f^{k},p\rangle/d^{k}}$$=V_{\mu_{f}}$ (1)

on

C. Otherwise(1)holds

on

$\hat{\mathbb{C}}-\bigcup_{k>0}f^{-k}(p)$.

We also characterize such pointsthat thepotentials actuallyconverge there. MainTheorem

3

(convergenceof potentials andpointwisebehavior). Let

_f

$\in$

Ratbe

_of

degree$d\geq 2$. For$p\in\hat{\mathbb{C}}-E(f)$ and$q\in\hat{\mathbb{C}}$,

$\lim_{karrow\infty}V_{\mu^{(}f^{k},p)/d^{\mathrm{A}}}(q)=V_{\mu_{f}}(q)$ (2)

if

and only

_if

$\lim_{karrow\infty}\frac{1}{d^{k}}\log\frac{1}{[p,f^{k}(q)]}=0$. (3)

ACKNOWLEDGMENT. This work

was

partially done while the author

was

a

long term researcher of International Project Research

2003

“Complex Dynam-ics” of RIMS ofKyoto University. The authoris

very

grateful toProf. Mitsuhiro

Shishikura, who is thechairofthe project, andthe staff of RIMS of Kyoto Uni-versity for theirhospitality,

Hewouldlike to

express

his gratitudetoProf. MasahikoTaniguchiandProf.Toshiyuki Sugawafor

many

invaluablediscussions and advices, toProf. Vincent Guedji for

helpful discussions, and to Prof. Peter HaissinskyandProf. Mitsuhiro Shishikura for usefulcomments

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