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Summary: On the dynamics of the renormalization operator
A. Avila, M. Martens, and W. de Melo 1
Abstract
An important part of the bifurcation diagram of unimodal maps
corresponds to in nite renormalizable maps. The dynamics of the
renormalization operator describes this part of the bifurcation pat-
tern precisely. Here we analyze the dynamics of the renormalization
operator acting on the space of C k in nitely renormalizable maps of
bounded type. We prove that two maps of the same type are expo-
nentially asymptotic. We suppose k 3 and quadratic critical point.
1 Introduction
We will consider the renormalization operator R acting on an open set in
the space of C k , k 3, of unimodal interval maps of bounded combinatorial
type. Each map f in the domain D of the operator has a periodic interval
around the critical point whose period q 2 is at most a given integer
N 3 and R(f) is aĈnely conjugate to the restriction of f q to this interval.
A map f is in nitely renormalizable of combinatorial type bounded by N
if all iterates R n (f) belong to D. Sullivan proved in [5], see also [3], the
existence of a compact invariant subset D such that the restriction
of R to is a homeomorphism which is topologically conjugate to a full
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