Deformations of group actions

First cohomology, rigidity and deformations of isometric group actions

There is an error in the proof of Theorem 1.1 that invalidates proofs of other theorems. Theorem 1.5 is unaffected.

Proper affine actions and geodesic flows of hyperbolic surfaces

We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomology class, this map is uniformly positive or uniformly negative on the space of geodesic currents if and only if the corresponding affine deformation is proper. As a corollary, the deformation space of proper affine deformations is an open convex cone.

李群胚作用的轨道固定形变空间

The orbit-fixing deformation spaces of an action of a Lie groupoid

本文通过模仿曲面Teichmüller空间重新表述了李群胚作用的形变空间，推广至可能非局部自由的作用与非紧流形，证明了上链刚性意味着形变空间为单点，并计算了$\mathop{\mathrm{PSL}}(2,\mathbb{R})$的具体形变空间。

标准映射的变形及其动作与李亚普诺夫指数

Deformations of the Standard Map with Prescribed Actions and Lyapunov Exponents

本文研究了标准映射的非平凡变形，保持无限多个周期轨道的辛动作和李亚普诺夫指数，并通过共振正则形和压缩映射原理实现了指定谱数据的变形。

Integrable deformations of Rössler and Lorenz systems from Poisson-Lie groups

A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrab…

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.