Convergence groups and semi conjugacy

Boundary actions of lattices and $C^0$ local semi-rigidity

We consider actions of cocompact lattices in semisimple Lie groups of the noncompact type on their boundaries $G/Q$, $Q$ a parabolic group, the so-called standard actions. We show that perturbations of the standard action in the homeomorphism group continuously factor onto the original standard action by a semi-conjugacy close to the identity. This generalizes works by Bowden, Mann, Manning and Weisman in the setting of negative curvature or Gromov hyperbolic groups. Finally, we also construct perturbations of the action of lattices on the geodesic boundary which are not $C^0$ semi-conjugate t…

Applications of the Brauer complex: card shuffling, permutation statistics, and dynamical systems

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type $A$ are found. This leads to new models of card shuffling and has interesting combinatorial and numb…

A classification of degree $2$ semi-stable rational maps $\mathbb{P}^2\to\mathbb{P}^2$ with large finite dynamical automorphism group

Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\mathbb P}^2_K\dashrightarrow{\mathbb P}^2_K$ whose automorphism group $$\text{Aut}(f):=\{ϕ\in\text{PGL}_3(K): ϕ^{-1}\circ f\circϕ=f\}$$ is finite and of order at least $3$. In particular, we prove that $\#\text{Aut}(f)\le24$ in general, that $\#\text{Aut}(f)\le21$ for morphisms, and that $\#\text{Aut}(f)\le6$ for all but finitely many conjugacy classes of $f$.

generalized区间交换变换与其逆变换在冻结时正则性之间的意外差异

A surprising discrepancy in the regularity of conjugacies between generalized interval exchange transformations and their inverses at freezing

研究揭示了广义区间交换变换与其逆在冻结时正则性的不对称行为，通过热力学正则化和零温极限分析，获得Hausdorff维度和Hölder指数的渐进行为。

Exponential Mixing and Smooth Classification of Commuting Expanding Maps

We show that genuinely higher rank expanding actions of abelian semi-groups on compact manifolds are $C^{\infty}$-conjugate to affine actions on infra-nilmanifolds. This is based on the classification of expanding diffeomorphisms up to \holder conjugacy by Gromov and Shub, and is similar to recent work on smooth classification of higher rank Anosov actions on tori and nilmanifolds. To prove regularity of the conjugacy in the higher rank setting, we establish exponential mixing of solenoid actions induced from semi-group actions by nilmanifold endomorphisms, a result of independent interest. We…

Maximal Convergence Groups and Rank One Symmetric Spaces

We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Möbius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the $d$-sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.

Topological Classification of Holomorphic, Semi-Hyperbolic Germs, in "Quasi-Absence" of Resonances

The classification, by topological conjugacy, of invertible holomorphic germs $f:(\mathbb{C}^n,0)\to (\mathbb{C}^n,0)$, with $λ_1,...,λ_n$ eigenvalues of $df_0$, and $|λ_i|\neq 1$ for $i=2,...,n$ while $λ_1$ is a root of the unity, is given, in the suitable hypothesis of ``quasi-absence'' of resonances.

可解群与直线上的仿射作用

Solvable Groups and Affine Actions on the Line

本文研究有限生成可解群在实区间上的保向作用，证明C^1微分同胚作用必半共轭于仿射作用，并给出同胚情形下的局部半共轭结果及仿射作用族的鲁棒性条件。