Hyperbolic-parabolic deformations of rational maps

We develop a Thurston-like theory to characterize geometrically finite rational maps, then apply it to study pinching and plumbing deformations of rational maps. We show that in certain conditions the pinching path converges uniformly and the quasiconformal conjugacy converges uniformly to a semi-conjugacy from the original map to the limit. Conversely, every geometrically finite rational map with parabolic points is the landing point of a pinching path for any prescribed plumbing combinatorics.Comment: 78 pages, 6 figure

Teichmüller spaces and holomorphic dynamics

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

Wandering Julia components of cubic rational maps

We prove that every wandering Julia component of cubic rational maps eventually has at most two complementary components.Comment: 10 pages, 2 figure

Renormalization and wandering continua of rational maps

24 pagesRenormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent work of Cui-Tan shows that these maps can be decomposed into postcritically fnite renormalization pieces. The main purpose of the present work is to perform the surgery one step deeper. Based on Thurston's idea of decompositions along multicurves, we introduce a key notion of Cantor multicurves (a stable multicurve generating infnitely many homotopic curves under pullback), and prove that any postcritically fnite piece having a Cantor multicurve can be further decomposed into smaller postcritically fnite renormalization pieces. As a byproduct, we establish the presence of separating wandering continua in the corresponding Julia sets. Contrary to the polynomial case, we exploit tools beyond the category of analytic and quasiconformal maps, such as Rees-Shishikura's semi-conjugacy for topological branched coverings that are Thurston-equivalent to rational maps