Fixed Flowers

TL;DR

This paper explores the structure of Julia sets through laminations, focusing on fixed point portraits and their global counts, and investigates the relationship between specific types of laminations with rotational polygons.

Contribution

It introduces new insights into fixed point portraits in laminations and examines the correspondence between locally unicritical and maximally critical laminations with rotational polygons.

Findings

Global count of fixed point portraits established

Correspondence between locally unicritical and maximally critical laminations analyzed

Relationship with rotational polygons clarified

Abstract

Laminations are a combinatorial and topological way to study Julia sets. Laminations give information about the structure of parameter space of degree $d$ polynomials with connected Julia sets. We first study fixed point portraits in laminations and their respective global count. Then, we investigate the correspondence between locally unicritical laminations and locally maximally critical laminations with rotational polygons. The global correspondence has been shown in \cite{Burdette:2022}.