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Curve attractors for marked rational maps

Zachary Smith

Abstract

A Thurston map $f\colon (S^2, A) \to (S^2, A)$ with marking set $A$ induces a pullback relation on isotopy classes of Jordan curves in $(S^2, A)$. If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair $(f,A)$ is said to have a finite global curve attractor. It is conjectured by Pilgrim that all rational Thurston maps that are not flexible Lattès maps have a finite global curve attractor. We present partial progress on this problem. Specifically, we prove that if $A$ has four points and the postcritical set (which is a subset of $A$) has two or three points, then $(f,A)$ has a finite global curve attractor. We also discuss extensions of the main result to certain special cases where $f$ has four postcritical points and $A=P_f$. Additionally, we speculate on how some of these ideas might be used in the more general case.

Curve attractors for marked rational maps

Abstract

A Thurston map with marking set induces a pullback relation on isotopy classes of Jordan curves in . If every curve lands in a finite list of possible curve classes after iterating this pullback relation, then the pair is said to have a finite global curve attractor. It is conjectured by Pilgrim that all rational Thurston maps that are not flexible Lattès maps have a finite global curve attractor. We present partial progress on this problem. Specifically, we prove that if has four points and the postcritical set (which is a subset of ) has two or three points, then has a finite global curve attractor. We also discuss extensions of the main result to certain special cases where has four postcritical points and . Additionally, we speculate on how some of these ideas might be used in the more general case.
Paper Structure (20 sections, 29 theorems, 51 equations, 2 figures)

This paper contains 20 sections, 29 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Outline of Paper
  4. Preliminaries
  5. Notations and Conventions
  6. Basic Definitions
  7. Pullback Relation on Curves
  8. Mapping Class Groups
  9. The Moduli Space Map for $(f,A)$
  10. Teichmüller Space as a Cover of Moduli Space
  11. The Pullback Map for $(f,A)$
  12. The Global Curve Attractor for $(f,A)$
  13. Equivalence to Attractors on Cusps
  14. Horoballs
  15. Cusp Types
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $f\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a rational Thurston map with a set $A$ of four marked points. If the postcritical set $P_f\subseteq A$ has at most three points, then $(f,A)$ has a finite global curve attractor.

Figures (2)

  • Figure 1: Generic picture for $\delta>1$ case.
  • Figure 2: The map $(g,A)$ of Example \ref{['Lexample']}

Theorems & Definitions (69)

  • Conjecture 1.1: Global Curve Attractor Conjecture
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • ...and 59 more