Thurston's pullback map on the augmented Teichm\"uller space and applications

TL;DR

This paper extends Thurston's pullback map to the augmented Teichmüller space, providing new insights into its dynamics, boundary behavior, and applications to key conjectures and theorems in complex dynamics.

Contribution

It explicitly constructs the extension of Thurston's map to the augmented space and characterizes its boundary dynamics, leading to new proofs and classification results.

Findings

Extended Thurston's pullback map continuously to augmented space

Characterized boundary dynamics and invariant strata

Provided new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem

Abstract

Let $f$ be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map $\sigma_f$ of a finite-dimensional Teichm\"uller space. We prove that this map extends continuously to the augmented Teichm\"uller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichm\"uller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.