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RENORMALIZATION IN COMPLEX DYNAMICS (Applications of Renormalization Group Methods in Mathematical Sciences)

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RENORMALIZATION

IN

COMPLEX DYNAMICS

MITSUHIRO SHISHIKURA (KYOTO UNIVERSITY)

The renormalization has been

one

of important tools and objectives in the study of (low-dimensional) dynamical systems, since it

was

introduced by Feigenbaum and Coullet-Tresser.

Thier goal

was

to explainthe universality in the bifurcation phenomenaof families ofunimodal

mappings

on

the interval. For this

purpose,

they defined arenormalization “operator” (instead

ofgroup)

on

the space ofunimodal maps and hypothesized the existenceofits fixed point and the hyperbolicity of its derivative. This

was

proved by Lanford in

1982

by acomputer-assisted proof. In 1980’s, there

were

works towards

anon

computer-assisted proof, and this created

a

new movement in the studyoflow-dimensional dynamics.

In this talk,

we

discuss the relationship between the renormalization and the problem of rigidity. Therigidity

means

that with acertain class of mathematicalobjects, aweakequivalence automatically implies astronger equivalence. For example, in the

case

of

Feigenbaum-Coullet-Tresserrenormalization, Lanford’stheorem impliesthat two Feigenbaum renoramalaizablemaps

with certain smoothness

are

smoothly $(C^{1})$ conjugate

on

their limit

Cantor

sets. There

are

various questions related to the rigidityofreal

or

complex

one

dimensional dynamical systems. The main result

we

discuss will be

$\mathrm{T}\dot{\mathrm{h}}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{m}$

.

Let

$f$ and$g$ be polynomial-like mappings with the

same

renormalization type which is not satellite type. Then they

are

quasiconformally conjugate outside the renormalizing Yoccoz

puzzle piece and the quasiconformal dilatation depends only on the combinatorial type

of

the renoramalization and the moduli

of

the

fundamental

annuli

of

$f$ and$g$

.

Moreover

if

both$f$ and$g$ are $rwl$, the dilatation depends only onthe moduli

of

the

fundamental

annuli.

Applying this theorem to the seuqence of renormalizations,

we

obtain

anew

proofof the

following:

Theorem. Hyperbolic maps

are

dense among real quadratic polynomials

数理解析研究所講究録 1275 巻 2002 年 126-126

参照

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