On the pullback relation on curves induced by a Thurston map

TL;DR

This paper surveys the dynamics of the pullback relation on curves induced by Thurston maps, exploring how preimage components affect isotopy classes and posing open questions in the field.

Contribution

It provides a comprehensive overview of known results and highlights open problems regarding the pullback relation on curves induced by Thurston maps.

Findings

Summarizes existing results on the pullback relation dynamics.

Identifies key open questions in the study of Thurston map-induced relations.

Connects the pullback relation to the broader context of complex dynamics.

Abstract

Via taking connected components of preimages, a Thurston map $f: (S^2, P_f) \to (S^2, P_f)$ induces a pullback relation on the set of isotopy classes of curves in the complement of its postcritical set $P_f$. We survey known results about the dynamics of this relation, and pose some questions.