Boundary of the central hyperbolic component I: dynamical properties

TL;DR

This paper investigates the boundary dynamics of polynomial maps within the central hyperbolic component, establishing local connectivity of Julia sets and a rigidity theorem using puzzle techniques and Fatou trees.

Contribution

It introduces new methods for analyzing boundary dynamics, proving local connectivity and rigidity for a broad class of polynomial maps.

Findings

Julia sets are locally connected on the boundary of the hyperbolic component

A rigidity theorem is established for maps on the regular boundary

The methods apply to maps with maximal Fatou trees equal to filled Julia sets

Abstract

We study the dynamics of polynomial maps on the boundary of the central hyperbolic component $\mathcal H_d$. We prove the local connectivity of Julia sets and a rigidity theorem for maps on the regular part of $\partial\mathcal H_d$. Our proof is based on the construction of Fatou trees and employs the puzzle technique as a key methodological framework. These results are applicable to a larger class of maps for which the maximal Fatou trees equal the filled Julia sets.