Hausdorff dimension and conformal measures of Feigenbaum Julia sets

Abstract

1.1. Statement of the results. One of the first questions usually asked about a fractal subset of R is whether it has the maximal possible Hausdorff dimension, n. It certainly happens if the set has positive Lebesgue measure. On the other hand, it is easy to construct fractal sets of zero measure but of dimension n. Moreover, this phenomenon is often observable for fractal sets produced by conformal dynamical systems, iterated rational functions or Kleinian groups. In particular, the analogy with Kleinian groups suggested that the Julia sets of Feigenbaum maps should have Hausdorff dimension two. In this paper we will show that this is not always the case. A quadratic polynomial (or more generally, a quadratic-like map) is called Feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds (see §2 for a precise definition). Feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years. By analogy with Kleinian groups, it was anticipated that their Julia sets have dimension 2, and this problem has been around for a while. However, in this paper we will show that this is not always the case: