Parabolic and near-parabolic renormalizations for local degree three

TL;DR

This paper introduces a new invariant class of holomorphic maps with degree three under parabolic and near-parabolic renormalizations, extending recent progress from quadratic to cubic cases in complex dynamics.

Contribution

It constructs a novel invariant class for degree three maps and explores potential applications to cubic unicritical polynomial dynamics.

Findings

Existence of cubic unicritical Julia sets with positive area

Characterizations of topology and geometry of cubic irrationally indifferent attractors

Extension of quadratic case results to cubic case

Abstract

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