Non-degenerate near-parabolic renormalization

TL;DR

This paper extends the concept of parabolic and near-parabolic renormalization to a more general setting, allowing for invariant classes when the multiplier is near multiple roots of unity, thus broadening the understanding of polynomial dynamics.

Contribution

It introduces a general framework for parabolic and near-parabolic renormalization applicable to multipliers near any root of unity, and relates multiple near-parabolic renormalizations.

Findings

Constructed invariant classes in the general setting.

Observed phenomena with multipliers near multiple roots of unity.

Established relations between different near-parabolic renormalizations.

Abstract

Invariant classes under parabolic and near-parabolic renormalization have proved extremely useful for studying the dynamics of polynomials. The first such class was introduced by Inou-Shishikura to study quadratic polynomials; their argument has been extended to the unicritical cubic case by Yang and the general unicritical case by Ch\'eritat. However, all of these classes are only applicable to maps which have a fixed point with multiplier close to one, though it is well-known that similar phenomena occur when the multiplier is close to any root of unity. In this paper we define the parabolic and near-parabolic renormalization operators in the general setting and construct invariant classes. In the general setting we can observe a new phenomenon: the multiplier may be close to several roots unity. In this case, we show how to directly relate the different near-parabolic renormalizations that arise.