Teichmüller spaces and holomorphic dynamics

Abstract

One fundamental theorem in the theory of holomorphic dynamics is Thurston's topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston's theorem (the marked Thurston's theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein.

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

“…6 we define the extension to the boundary of the augmented Teichmüller space explicitly in a way that is similar to the definition of the action of Thurston's pullback map on the Teichmüller space. This brings, in our opinion, new insights in understanding the behavior of Thurston's pullback map.…”

Hyperbolic-parabolic deformations of rational maps

“…This definition coincides with the definition in [3] of combinatorial equivalence when P is finite. Refer to [14,31] for the definition of hyperbolic orbifold and [3] for the following theorem. • f is post-critically finite and the signature of the orbifold of f is (2, 2, 2, 2).…”

“…In this section, we will prove Theorem 1. 3. Let f be a geometrically finite rational map and let A ⊂ R f be a starlike multi-annulus.…”

“…Let {U y } be a pullback system of Y as defined in §8. 3. Set n(y) = 0 if y ∈ Y 0 and n(y) = n if g n (y) ∈ Y 0 but g n−1 (y) ∈ Y 0 .…”

“…Because s 1 (r) → 0 as r → 0, there exists a constant r 4 ∈ (0, r 3 ) such that s 1 (r 4 ) < r 3 . For any two points x, y ∈ X with n(y) > n(x) ≥ n, if D y ∩ D x = ∅, then D y ⊂ D x by Property (2).…”