On the Lebesgue measure of the Feigenbaum Julia set

Abstract

A. We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than 2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.

On Lebesgue measure and Hausdorff dimension of Julia sets of real periodic points of renormalization

“…Thus,η n is the probability that the orbit of a point randomly chosen from W (1) with respect to Lebesgue measure will intersect W (n) . By construction, X n+1 ⊂ X n for any n. Therefore,η n is non-increasing in n. Similarly to [8] we have:…”

“…For instance, an open question is whether there exist quadratic Feigenbaum maps with real coefficients having Julia sets of positive area, or at least Hausdorff dimension 2. Jointly with Sutherland, the author of the present paper showed in [8] that the Julia set of the original quadratic Feigenbaum map (corresponding to the period-doubling renormalization) has Hausdorff dimension less than 2, and thus has zero area.…”

“…In the proof of Theorem 4 we closely follow the approach of [8]. However, we have to make several modifications compared to the case p = 2 studied in [8].…”

“…Here we generalize the approach of [8] to an arbitrary real periodic point of renormalization. Given such a map f , the main result of this paper (Theorem 4) provides a sufficient condition for the Julia set of f to be of Hausdorff dimension less than 2.…”

“…In [8] we showed with Sutherland that the Julia set of F * has Hausdorff dimension less than 2. The present paper concerns the measure and Hausdorff dimension of the Julia sets of the real periodic points of renormalization, i.e.…”