University of Rhode Island DigitalCommons@URI Open Access Dissertations 2018 Identifications in Escape Regions of the arP ameter Space of Cubic Polynomials Chad Estabrooks University of Rhode Island, [email protected] Follow this and additional works at: https://digitalcommons.uri.edu/oa_diss Recommended Citation Estabrooks, Chad, "Identifications in Escape Regions of the arP ameter Space of Cubic Polynomials" (2018). Open Access Dissertations. Paper 734. https://digitalcommons.uri.edu/oa_diss/734 This Dissertation is brought to you for free and open access by DigitalCommons@URI. It has been accepted for inclusion in Open Access Dissertations by an authorized administrator of DigitalCommons@URI. For more information, please contact [email protected]. IDENTIFICATIONS IN ESCAPE REGIONS OF THE PARAMETER SPACE OF CUBIC POLYNOMIALS BY CHAD ESTABROOKS A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS UNIVERSITY OF RHODE ISLAND 2018 DOCTOR OF PHILOSOPHY DISSERTATION OF CHAD ESTABROOKS APPROVED: Dissertation Committee: Major Professor Araceli Bonifant Thomas Sharland Mark Comerford G. Faye Boudreaux-Bartels Nasser H. Zawia DEAN OF THE GRADUATE SCHOOL UNIVERSITY OF RHODE ISLAND 2018 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 7! z2 − 1. ACKNOWLEDGMENTS I would like to first thank my advisors, Dr. Araceli Bonifant and Dr. Thomas Sharland. I can't imagine having been in better hands throughout the research process. I am extremely grateful for the time and effort that each of them has put into my education. It was extremely helpful that their office doors were often open for me to bring thoughts and questions for deep unplanned discussions. I also thank those who have inspired me in a particular way throughout my time studying mathematics. Dr. Christian Yankov and Dr. Steve Kenton were instrumental in showing me how exciting and beautiful mathematics can be. Dr. Rita Hibschweiler challenged me like I had never been challenged before. This taught me humility and helped me build a stronger work ethic. The encouragement of CAPT Melinda McGurer played a large role in my decision to continue my education. The passion and clarity of Dr. Lubos Thoma, and the rigor and attention to detail of Dr. Mark Comerford were much appreciated. I have been very blessed to be able to come home after long days of proving theorems to my amazing wife Sara, and three wonderful children; Eli, Lexi, and Faith. No matter how difficult things became, they were always able to brighten my day, and help to keep me focused. We are also very excited to welcome baby Cyril into the world during the final stages of putting together this work. I could never have come as far as I have without a lifetime of careful guidance. I would like to thank my parents, James and Lisa Estabrooks, for constantly, but lovingly, pushing me to strive to be the best that I can be. I am grateful to have always had their support through my failures as well as my successes. My parents-in-law, Sean and Laura Hanrahan, have provided tremendous sup- port in every possible way while I have pursued my education. It would not have been feasible to return to a doctoral program without this support. iii DEDICATION This work is dedicated to my amazing wife, Sara. Her love and support has been essential to keeping me focused and grounded. She has cheerfully and heroically made great sacrifices in order for me to be able to pursue my education. iv TABLE OF CONTENTS ABSTRACT .................................. ii ACKNOWLEDGMENTS .......................... iii DEDICATION ................................. iv TABLE OF CONTENTS ..........................v LIST OF FIGURES .............................. vii CHAPTER 1 Introduction ...............................1 List of References . .7 2 Preliminaries ..............................8 2.1 Dynamical Systems and Conjugacy . .8 2.2 Periodic Points and the Multiplier . .9 2.3 Dynamics of Rational Maps and Polynomials . 10 2.4 B¨ottcher's Theorem and Rays . 13 2.5 Cubic Polynomials . 18 2.6 Milnor's Curves in Cubic Parameter Space . 19 2.7 Escape Regions . 25 2.8 The Period q Decomposition of Sn ................. 28 2.9 A Conformal Isomorphism . 29 List of References . 31 3 The basilica map ............................ 32 v Page 3.1 Hubbard trees . 32 3.2 Laminations . 35 3.3 Hybrid equivalence . 40 List of References . 43 4 Identifications in Escape Regions .................. 45 4.1 Preliminary Results . 45 4.2 Behavior of the Conformal Isomorphism . 47 4.3 Main results . 59 List of References . 66 APPENDIX Table of Notation ............................. 67 BIBLIOGRAPHY ............................... 69 vi LIST OF FIGURES Figure Page 1 An image in the parameter space of quadratic polynomials, known as the Mandelbrot set. .2 2 A well known quadratic Julia set known as the Basilica, corre- sponding to the map z 7! z2 − 1. .3 3 Connectedness loci of the curves S1 and S2 (left and right, re- spectively) shown side by side. .4 4 The dynamics of the map z 7! z2.................. 11 5 The filled Julia set of the map z 7! z2 − 1, called the basilica. 14 6 Rational external rays of denominator eight for a quadratic poly- nomial known as Douady's rabbit. 16 7 A cubic polynomial with superattracting fixed point at a = 0:78i, and some internal and external rays marked. 17 8 A projection into one complex dimension of the curve S1..... 20 9 A projection into one complex dimension of the curve S2..... 21 10 Illustrations which demonstrate the interactions between the two critical points in the four different types of components. 23 11 A piece of C(S2) with components of all four types labeled. 24 12 An illustration of a typical boundary of the maximal domain of definition of the B¨ottcher coordinate for a map on the external parameter ray of argument t, with important dynamical external rays marked. 26 13 A region of S1 showing type C components homeomorphic to discs, and the corresponding image under Ψb in S2, showing type C components homeomorphic to the basilica. 29 14 Magnifications of the areas indicated in Figure 13 with some external rays marked . 29 vii Figure Page 15 Another region of S1 and the corresponding image under Ψb in S2. 30 16 Magnifications of the areas indicated in Figure 15 with some external rays marked. 30 17 Hyperbolic geodesics in the standard disk model. 33 18 The basilica with its Hubbard tree highlighted. 34 19 Douady's rabbit with its Hubbard tree highlighted. This tree requires four vertices, despite the map having three total critical and post-critical points. 35 20 The first four steps of the inductive process for obtaining the lamination of the basilica. 37 21 The lamination of the basilica. 38 22 A map on the left with some identifications from the lamination of the basilica shown in a disc component of the filled Julia set, and the result on the right of the quotient by the basilica lamination. 39 B 23 The filled Julia set of a map in E2 showing the placement of the fixed point α............................. 42 B 24 An updated version of Figure 12 for a map in E2 . The same applies to a map in E1 with a disc in place of the main basilica component illustrated. 49 25 S2 with the basilica escape region decomposed into regions ac- cording to the number of rays landing at the fixed point α.... 50 26 The pairs of rays landing at α in each piece of the basilica escape region. 52 27 The Julia set of the map at the center of the period two hyper- bolic component contained in the same component of the period two decomposition of S2 as the set W1, with period two external rays which land together marked. 53 28 A Julia set from W1 showing the four rays landing at the fixed point α................................ 55 viii Figure Page 29 The Julia set of a map from the component of the period two decomposition of S1 which contains the entire preimage under Ψ of W1 ⊂ S2............................. 57 30 The principal hyperbolic component of S1 split up into four quadrants. 63 31 An illustration of the contradiction from the proof of Theorem 4.18 when F is taken in the boundary of quadrant I of Hb0.... 65 ix CHAPTER 1 Introduction One of the major areas of holomorphic dynamics is the study of the iteration of holomorphic maps on the Riemann sphere Cb = C [ f1g. It is well known that any holomorphic map of the sphere can be expressed as a rational function. Polynomials are also widely studied as a special case of the study of rational maps. 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. These sets came to be known as the Fatou set, where the dynamics are tame, and the Julia set, where the dynamics are chaotic. After this period of work, the subject was largely forgotten until the 1980's when Adrien Douady and John Hubbard proved many important results, see [1].