Measure of the Julia Set of the Feigenbaum map with infinite criticality

TL;DR

This paper investigates the Lebesgue measure of Julia sets associated with Feigenbaum maps of infinite criticality, showing it tends to zero despite their hyperbolic dimension approaching 2, using martingale theory.

Contribution

It introduces a novel application of martingale theory to analyze the Lebesgue measure of Julia sets in the context of Feigenbaum maps with infinite criticality.

Findings

Lebesgue measure of Julia sets tends to zero

Hyperbolic dimension approaches 2

Martingale theory applied to non-integrable increments

Abstract

We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.