Abstract

We investigate a relationship between escape regions in slices of the parameter space of cubic polynomials. The focus of this work is to give a precise description of how to obtain a topological model for the boundary of an escape region in the slice consisting of all cubic polynomials with a marked critical point belonging to a two cycle. To obtain this model, we start with the unique escape region in the slice consisting of all maps with a fixed marked critical point, and make identifications which are described using the identifications which are made in the lamination of the basilica map z → z2 – 1.

Highlights

  • One of the major areas of holomorphic dynamics is the study of the iteration of holomorphic maps on the Riemann sphere C = C ∪ {∞}

  • It is well known that any holomorphic map of the sphere can be expressed as a rational function

  • The field began in the 1920’s with two French mathematicians, Pierre Fatou and Gaston Julia. They showed that given a rational map, the Riemann sphere can be split into two disjoint sets

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Summary

Introduction

Conformally conjugate to a unique polynomial of this form He defined, for each natural number n, the curve Sn as the space consisting of all of the maps Pa,v for which the critical point a has period exactly n. In Theorem 4.16, we consider first a single type C component of S1 Such a component is necessarily a topological disc, and we are able to parametrize its boundary by the argument t ∈ R/Z of the internal parameter ray in this component landing at each point. We obtain a similar result here, there is an extra complication The boundary of this component is a topological circle, we have a two-fold covering map from ∂H0 to R/Z when associating each point on the boundary with the argument of the internal ray landing there.

CHAPTER 2 Preliminaries
Dynamical Systems and Conjugacy We begin with the basics of general dynamical systems
Periodic Points and the Multiplier Consider a polynomial p on C
Bottcher’s Theorem and Rays We will make use of the following theorem of
Cubic Polynomials We will now begin to focus our study on cubic polynomials
Escape Regions We now look at the subset of hyperbolic maps in
The Period q Decomposition of Sn
Laminations Consider a polynomial p whose Julia set is connected and locally connected
Hybrid equivalence
CHAPTER 4 Identifications in Escape Regions
Behavior of the Conformal Isomorphism
Main results We begin with our main results by extending the result of

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