 
Summary: EXAMPLES OF FEIGENBAUM JULIA SETS
WITH SMALL HAUSDORFF DIMENSION
ARTUR AVILA AND MIKHAIL LYUBICH
To Bodil Branner on her 60th birthday
Abstract. We give examples of infinitely renormalizable quadratic polynomials Fc : z z2 + c
with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrary close to 1. The
combinatorics of the renormalization involved is close to the Chebyshev one. The argument is
based upon a new tool, a "Recursive Quadratic Estimate" for the Poincar´e series of an infinitely
renormalizable map.
1. Introduction
One of the most remarkable objects in complex dynamics are the fixed points of the Douady
Hubbard renormalization operator. Such objects have a distinguished place in the dictionary be
tween rational maps and Kleinian groups (see [Mc2]). Existence of the renormalization fixed points
established in the works of Sullivan [S2] and McMullen [Mc2] (under certain assumptions) implies
many beautiful features (selfsimilarity, universality, hairyness,...) of Feigenbaum Julia sets (see §2.2
for the definition). However, even with this thorough information, some basic questions concerning
measure and dimension of these Julia sets have remained unsettled.
One of the key questions (asked, for instance, in [Mc2]) regarding the geometry of Feigenbaum
Julia sets has been the following: Is the Hausdorff dimension of a Feigenbaum Julia set always equal
to 2? In [AL] we supply a fairly large class of Feigenbaum Julia sets with HD(J) < 2, thus giving
