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 itwas
introduced by Feigenbaum and Coullet-Tresser.Thier goal
was
to explainthe universality in the bifurcation phenomenaof families ofunimodalmappings
on
the interval. For thispurpose,
they defined arenormalization “operator” (insteadofgroup)
on
the space ofunimodal maps and hypothesized the existenceofits fixed point and the hyperbolicity of its derivative. Thiswas
proved by Lanford in1982
by acomputer-assisted proof. In 1980’s, therewere
works towardsanon
computer-assisted proof, and this createda
new movement in the studyoflow-dimensional dynamics.
In this talk,
we
discuss the relationship between the renormalization and the problem of rigidity. Therigiditymeans
that with acertain class of mathematicalobjects, aweakequivalence automatically implies astronger equivalence. For example, in thecase
ofFeigenbaum-Coullet-Tresserrenormalization, Lanford’stheorem impliesthat two Feigenbaum renoramalaizablemaps
with certain smoothness
are
smoothly $(C^{1})$ conjugateon
their limitCantor
sets. Thereare
various questions related to the rigidityofreal
or
complexone
dimensional dynamical systems. The main resultwe
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 theyare
quasiconformally conjugate outside the renormalizing Yoccozpuzzle piece and the quasiconformal dilatation depends only on the combinatorial type
of
the renoramalization and the moduliof
thefundamental
annuliof
$f$ and$g$.
Moreover
if
both$f$ and$g$ are $rwl$, the dilatation depends only onthe moduliof
thefundamental
annuli.
Applying this theorem to the seuqence of renormalizations,
we
obtainanew
proofof thefollowing:
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