Typical orbits of quadratic polynomials with a neutral fixed point: Brjuno type

TL;DR

This paper investigates the topological and measure-theoretic properties of typical orbits of complex quadratic polynomials with a neutral fixed point, demonstrating that for Brjuno alpha values, the critical orbit closure has zero area and characterizing the Julia set limit set.

Contribution

It establishes that for high return Brjuno alpha, the critical orbit closure has zero area and links the Julia set's typical orbit limit set to the critical orbit closure.

Findings

Critical orbit closure has zero area for Brjuno alpha.

The limit set of a typical Julia set orbit equals the critical orbit closure.

Results apply to quadratic polynomials with neutral fixed points.

Abstract

We describe the topological behavior of typical orbits of complex quadratic polynomials P_alpha(z)=e^{2\pi i alpha} z+z^2, with alpha of high return type. Here we prove that for such Brjuno values of alpha the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then combining with Part I of this work, we show that the limit set of the orbit of a typical point in the Julia set is equal to the closure of the critical orbit.