Thurston’s pullback map on the augmented Teichmüller space and applications

Abstract

Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller 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 Teichmüller 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.

Nearly Euclidean Thurston maps

“…We define the modulus of this family of curves just as we defined mod τ ( p q ). Because lengths of curves do not change when pulling back from S τ to T τ but area doubles, this new modulus is [10] shows that Σ f is deg(f )-Lipschitz with respect to the WP metric, and so extends to the WP completion. It easily follows that for NET maps, computation of σ f is the computation of the boundary values of Σ f on the WP completion of Teichmüller space.…”

Origami, Affine Maps, and Complex Dynamics

“…(5) The homotopy classes of curves in S 2 − P (f ) are classified by their slopes, that is, elements of Q = Q∪{±1/0 = ∞}; with conventional identifications, each slope p/q corresponds to the ideal boundary point −q/p ∈ ∂H. (6) By taking preimages of curves, we obtain a slope function µ f : Q → Q∪{ } where denotes the union of inessential and peripheral homotopy classes; this encodes the Weil-Petersson boundary values of σ f , shown to exist in general by Selinger [26]. More precisely, if p…”

Algorithmic aspects of branched coverings IV/V. Expanding maps

“…Theorem 4.10 (Pilgrim, Selinger [19]). Every small Thurston map in the canonical decomposition of f is either ‚ combinatorially equivalent to a rational non-Lattes post-critically finite map; ‚ double covered by a torus endomorphism; ‚ or a homeomorphism.…”