Pullback invariants of Thurston maps

Boundary Values of the Thurston Pullback Map

For any Thurston map with exactly four postcritical points, we present an algorithm to compute the Weil-Petersson boundary values of the corresponding Thurston pullback map. This procedure is carried out for the Thurston map f(z) = 3z 2 2z3+1 originally studied by Buff, et al. The dynamics of this boundary map are investigated and used to solve the analogue of Hubbard’s Twisted Rabbit problem f...

Nearly Euclidean Thurston Maps

We take an in-depth look at Thurston’s combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S2 → S2 whose local degree at every critical point is 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to ha...

On Nearly Euclidean Thurston Maps

Pullback of Varieties by Finite Maps

We study the local geometry of the pullback of a variety via a finite holomorphic map. In particular, we are looking for properties of V = F(W ) such that if V has the property A, then W must have the property A. We show that A can be the property of normality or prefactoriality. We also show that A can be the property of smoothness, under extra assumptions.

Invariant Peano Curves of Expanding Thurston Maps

We consider Thurston maps, i.e., branched covering maps f : S → S that are postcritically finite. It is shown that a Thurston map f is expanding (in a suitable sense) if and only if some iterate F = f is semi-conjugate to z : S → S, where d = deg F . More precisely, for such an F we construct a Peano curve γ : S → S (onto), such that F ◦ γ(z) = γ(z) (for all z ∈ S).