Symmetric Laminations of Degree d and Identity Return Triangles in Degree 3

Authors

• Thomas Sirna, University of Alabama at Birmingham

Advisor(s)

John Mayer
Lex Oversteegen

Committee Member(s)

Alexander Blokh
Bulent Tosun
Dongsheng Wu
Thomas Gilray

School

College of Arts and Sciences

Document Type

Dissertation

Department (new version)

Applied Mathematics

Date of Award

9-9-2024

Abstract

Laminations of the unit disc were introduced by Thurston in the 1980s as a tool to study the Julia sets and the parameter spaces of complex polynomials. In general, a lamination will be the unit disc D (considered as a subset of the complex plane C, with the boundary of D denoted by S) along with a collection of chords in D that can only intersect at their endpoints on S. We call the chords of a lamination leaves. In most situations we also consider a map σd : S → S on the endpoints of the leaves defined by σd(z) = zd, and laminations that satisfy certain invariant properties with respect to σd are called laminations of degree d. While giving the introductory definitions and results we will give an emphasis to finite gaps of laminations — in particular to triangular gaps — and to what they correspond to in a complex polynomial’s Julia set. This is because many of the main problems of this dissertation directly involve questions about triangular gaps. Laminations of degree d that are invariant under a rotation that depends on d are called symmetric laminations. In particular symmetric laminations of degree 3 have a 180à symmetry and have been studied previously. We extend most of the results to degree d with the appropriate generalizations, the first of which is that they exhibit a 360à/(d-1) rotational symmetry. Two of the main results for symmetric laminations will be that triangular gaps cannot have infinite forward orbit under σd (No Wandering Triangles), but that, contrary to the degree 3 case, a type of finite gap called identity return triangles can exist for specific higher degrees. Then we focus on just degree 3 laminations in general and demonstrate a correspon- dence between certain periodic leaves that we call 2 × 2 base leaves (sometimes just ii called “base leaves”) and certain periodic triangular gaps that we call multi critical moment identity return triangles (abbreviated MCM IRTs) by associating to 2×2 base leaves specific σ3-periodic points on S that we call co-roots. This is complementary to previous work done for single critical moment identity return triangles (SCM IRTs) and together gives a full picture of periodic triangular gaps in laminations of degree 3. In particular we look for a characterization of these base leaves and for a constructive way to associate them with their co-roots to form the appropriate MCM IRTs. At the end we use our results to give the first two generalizations of these ideas to polygonal gaps in σd. Next we will exhibit and explain some code written in Racket by the author in order to first automate a search for periodic leaves for any σd by creating a file of basic laminational procedures. Then this basic set of procedures is used in another file to develop an algorithm which generates all 2 × 2 base leaves in degree 3 of a given period by using details of the above results. Once a comprehensive list of these base leaves are generated one can use the written procedures to find their co-roots and their associated MCM IRTs. Finally a gallery of pictures of all 2 × 2 base leaves in σ3 up to a certain period — and some other examples produced by the Racket code — is given in the appendix as a comprehensive reference of examples, which we consider extremely useful considering that up until this point we have only been able to make use of a handful of examples of small period in order to formulate and test any conjectures. Along with a gallery of these laminations we also give some tables of data produced by the code for easy reference.

ProQuest ID

https://proquest.com/docview/3101880712