On Thurston's pullback map

TL;DR

This paper explores explicit examples of Thurston's pullback map for rational functions, illustrating diverse dynamical behaviors including attracting, superattracting, and constant cases, thereby enriching understanding of its complex dynamics.

Contribution

It provides explicit examples demonstrating various possible behaviors of Thurston's pullback map, including attracting, surjective, and constant cases, which were previously not fully illustrated.

Findings

The pullback map can have an attracting fixed point with dense image.

It can be a surjective ramified Galois covering.

In some cases, the pullback map is constant.

Abstract

Let f: P^1 \to P^1 be a rational map with finite postcritical set P_f. Thurston showed that f induces a holomorphic map \sigma_f of the Teichmueller space T modelled on P_f to itself fixing the basepoint corresponding to the identity map (P^1, P_f) \to (P^1, P_f). We give explicit examples of such maps f showing that the following cases may occur: (1) the basepoint is an attracting fixed point, the image of \sigma_f is open and dense, and the map \sigma_f is a covering map onto its image; (2) the basepoint is a superattracting fixed point, \sigma is surjective, and \sigma is a ramified Galois covering, (3) \sigma_f is constant.