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In a previous paper, we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces., Comment: 56 pages, 13 figures. Final version, to appear in Geom. Funct. Anal

In this paper, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps., 49 pages, 14 figures. The last two sections of v1 are moved to the sequel of this paper. Final version, to appear in Forum of Math. Sigma

In this paper, we develop a theory on the degenerations of Blaschke products $\mathcal{B}_d$ to study the boundaries of hyperbolic components. We give a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component $\mathcal{H}_d$ containing $z^d$. We show the closure $\overline{\mathcal{H}_d}$ is not a topological manifold with boundary for $d\geq 4$ by constructing self-bumps on its boundary., 68 pages, 34 figures. Final version, to appear in J. Eur. Math. Soc. (JEMS)

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence of rational maps of a fixed degree with infinitely many non-monomial rescaling limits., Comment: 45 pages, 26 figures. Final version, to appear in J. Lond. Math. Soc

In this paper, we show that one can naturally associate a limiting dynamical system $F: T\longrightarrow T$ on an $\R$-tree to any degenerating sequence of rational maps $f_n: \hat\C \longrightarrow \hat\C$ of fixed degree. The construction of $F$ is in $2$ steps: first we use barycentric extension to get $\E f_n : \Hyp^3 \longrightarrow \Hyp^3$; second, we take appropriate limit on rescalings of hyperbolic space. An important ingredient we prove here is that the Lipschitz constant depends only on the degree of the rational map. We show that the dynamics of $F$ records the limiting length spectra of the sequence $f_n$., Comment: 41 pages, 5 figure. Final version