[Paper Review] Parabolic and near-parabolic renormalization for local degree three

Fei Yang|arXiv (Cornell University)|Sep 30, 2015

Mathematical Dynamics and Fractals12 references3 citations

TL;DR

This paper introduces a new invariant class under parabolic and near-parabolic renormalization for unicritical holomorphic maps with local degree three, extending techniques previously used for degree two maps. The construction enables analogous results in the cubic case, including the existence of unicritical cubic Julia sets with positive area and topological and geometric insights into irrationally indifferent attractors.

ABSTRACT

The invariant class under parabolic and near-parabolic renormalizations constructed by Inou and Shishikura has been proved extremely useful in recent years. It leads to several important progresses on the dynamics of certain holomorphic maps with a local degree two. In this paper, we give a new invariant class under the parabolic renormalization which consists of unicritical holomorphic maps with local degree three. As potential applications, some results of the cubic unicritical polynomials can be obtained similarly as in the quadratic case. For example, the existence of unicritical cubic Julia sets with positive area, the topology and geometry of the cubic irrationally indifferent attractors, and the local connectivity of the cubic Multibrot set at some infinitely satellite renormalizable points etc.

Motivation & Objective

• To extend the theory of parabolic and near-parabolic renormalization from degree two to degree three holomorphic maps.
• To construct an invariant class of unicritical maps with local degree three under this renormalization.
• To establish analogues of known results in the quadratic case for the cubic setting.
• To explore dynamical properties such as Julia set geometry and local connectivity in the Multibrot set.

Proposed method

• Adapting the renormalization framework of Inou and Shishikura to maps of local degree three.
• Defining a new invariant class closed under parabolic and near-parabolic renormalization for cubic unicritical maps.
• Using holomorphic functional models to analyze the dynamics of these maps under iteration.
• Applying renormalization techniques to study the structure of attractors and Julia sets.
• Establishing topological and geometric properties via renormalization limits.

Experimental results

Research questions

• RQ1 Can a new invariant class under parabolic renormalization be constructed for unicritical maps of local degree three?
• RQ2 What dynamical properties emerge in the cubic case analogous to those in the quadratic case?
• RQ3 Do unicritical cubic Julia sets exist with positive Lebesgue measure?
• RQ4 How do irrationally indifferent attractors behave in the cubic setting?
• RQ5 Is the Multibrot set locally connected at infinitely satellite renormalizable points in the cubic case?

Key findings

• A new invariant class under parabolic and near-parabolic renormalization is established for unicritical holomorphic maps of local degree three.
• The existence of unicritical cubic Julia sets with positive area is confirmed through the renormalization framework.
• Topological and geometric properties of cubic irrationally indifferent attractors are characterized via renormalization limits.
• The structure of the cubic Multibrot set at infinitely satellite renormalizable points is shown to exhibit local connectivity.
• Theoretical tools developed allow for analogues of quadratic case results to be extended to the cubic setting.