On the rank of the Thurston pullback map

TL;DR

This paper investigates the rank of the Thurston pullback map induced by branched coverings of marked spheres, providing bounds and insights into postcritical set dynamics relevant to Thurston's characterization of rational maps.

Contribution

It introduces a method to analyze the pullback map's derivative rank via quadratic differentials, offering bounds on postcritical set size and restrictions on postcritical dynamics.

Findings

Lower bounds on the rank of the pullback map derivative.

Upper bounds on the size of the postcritical set for maps with constant pullback.

Restrictions on postcritical dynamics for certain branched coverings.

Abstract

Under some mild assumptions, an orientation-preserving branched covering map of marked $2$-spheres induces a pullback map between the corresponding Teichm\"uller spaces. By analyzing the associated pushforward operator acting on integrable quadratic differentials, we obtain a global lower bound on the rank of the derivative of the pullback map in terms of the action of the cover on the marked points. In the dynamical context, the two sets of marked points in the target and source coincide with the postcritical set. Investigating the resulting pullback map is the central part of Thurston's topological characterization of postcritically finite rational maps. Postcritically finite maps with constant pullback have been studied by various authors. In that direction, our approach provides upper bounds on the size of the postcritical set of a map with constant pullback, and shows that the postcritical dynamics is highly restricted.