 
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 innite 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 innitely 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 innitely 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
