Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Thurston's pullback map, invariant covers, and the global dynamics on curves

Mario Bonk, Mikhail Hlushchanka, Russell Lodge

Abstract

We consider rational maps $f$ on the Riemann sphere $\widehat {\mathbb{C}}$ with an $f$-invariant set $P\subset \widehat {\mathbb{C}}$ of four marked points containing the postcritical set of $f$. We show that the dynamics of the corresponding Thurston pullback map $σ_f$ on the completion $\overline{\mathcal{T}_P}$ of the associated Teichmüller space $\mathcal{T}_P$ with respect to the Weil-Petersson metric is easy to understand when $\overline{\mathcal{T}_P}$ admits a cover by sets with good combinatorial and dynamical properties. In particular, the map $f$ has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if $P$ contains all critical points of $f$ and each point in $P$ is periodic, then such a cover of $\overline{\mathcal{T}_P}$ can be obtained from a $σ_f$-invariant tessellation by ideal hyperbolic triangles.

Thurston's pullback map, invariant covers, and the global dynamics on curves

Abstract

We consider rational maps on the Riemann sphere with an -invariant set of four marked points containing the postcritical set of . We show that the dynamics of the corresponding Thurston pullback map on the completion of the associated Teichmüller space with respect to the Weil-Petersson metric is easy to understand when admits a cover by sets with good combinatorial and dynamical properties. In particular, the map has a finite global curve attractor in this case. Using a result by Eremenko and Gabrielov, we also show that if contains all critical points of and each point in is periodic, then such a cover of can be obtained from a -invariant tessellation by ideal hyperbolic triangles.

Paper Structure

This paper contains 7 sections, 6 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Thurston's pullback map
  3. Tile covers and global attractors
  4. Applications and examples
  5. Rational Thurston maps with marked critical points
  6. Rational Thurston maps with a moduli space map
  7. Examples

Key Result

Theorem 1

Let $f\colon (\widehat{\mathbb{C}}, P)\to (\widehat{\mathbb{C}}, P)$ be a rational Thurston map with $|P|=4$ and a hyperbolic orbifold. Suppose that there is a cover $\mathcal{U}$ of the Weil--Petersson completion $\overline{\mathcal{T}_P}=\mathbb{H}^*$ by some of its subsets such that the following Then the pullback relation $\xleftarrow{f}$ on curves has a finite global attractor.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • proof : Proof of Theorem \ref{['thm-intro:FGA-tiling']}
  • Claim
  • Lemma 3
  • proof
  • Lemma 4
  • proof : Proof of Theorem \ref{['thm:inv-tiling-crit-points']}
  • Remark
  • Corollary 5
  • ...and 1 more