Spinning deformations of rational maps

Non co-preservation of the $1/2$ & $1/(2l+1)$-rational caustics along deformations of circles

For any given positive integer $l$, we prove that every plane deformation of a circle which preserves the $1/2$ and $1/(2l+1)$-rational caustics is trivial i.e. the deformation consists only of similarities (rescalings plus isometries).

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.

二次有理映射的形变空间

On deformation spaces of quadratic rational maps

研究部分后临界有限分支自覆盖的自等价群，解答了关于有理映射形变空间可收缩性的问题。

Infinitesimal deformations of rational surface automorphisms

If $X$ is a rational surface without nonzero holomorphic vector field and $f$ is an automorphism of $X$, we study in several examples the Zariski tangent space of the local deformation space of the pair $(X, f)$.

The augmented deformation space of rational maps

The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, S. Koch and I recently showed that the deformation space has infinitely many connected components. In the present paper we study the augmented deformation space for this example, and show, in particular, that the closure of deformation space in augmented deformation space is also disconnected in this case.

On Deformation Space Analogies between Kleinian Reflection Groups and Antiholomorphic Rational Maps

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.