Dynamic Asymptotic Dimension of Translation Actions on Compact Lie Groups

关于平移流的$\Z^d$-扩张的遍历积分的渐近展开

On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows

本文研究紧自相似平移流的$\Z^d$-覆盖，建立了遍历积分的渐近展开，并由此得到具有最优速率的弱有理遍历性，同时应用于自相似阶梯流和Ehrenfest风树模型。

Cocycle superrigidity for translation actions of product groups

Let $G$ be either a profinite or a connected compact group, and $Γ, Λ$ be finitely generated dense subgroups. Assuming that the left translation action of $Γ$ on $G$ is strongly ergodic, we prove that any cocycle for the left-right translation action of $Γ\timesΛ$ on $G$ with values in a countable group is virtually cohomologous to a group homomorphism. Moreover, we prove that the same holds if $G$ is a (not necessarily compact) connected simple Lie group provided that $Λ$ contains an infinite cyclic subgroup with compact closure. We derive several applications to OE - and W$^*$- superrigidity…

关于球面复形上的渐近平移长度与广义纤维锥

On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone

本文研究圆周纤维化流形的幺模映射在球面复形上的渐近平移长度，定义广义纤维锥并证明其为Fried锥的有理切片，给出其真子锥上渐近平移长度的统一上界，并应用于柄体群在圆盘复形上的最小渐近平移长度。

The minimal Morse components of translations on flag manifolds are normally hyperbolic

Consider the iteration of an invertible matrix on the projective space: are the Morse components normally hyperbolic? As far as we know, this was only stablished when the matrix is diagonalizable over the complex numbers. In this article we prove that this is true in the far more general context of an arbitrary element of a semisimple Lie group acting on its generalized flag manifolds: the so called translations on flag manifolds. This context encompasses the iteration of an invertible non-diagonazible matrix on the real or complex projective space, the classical flag manifolds of real or comp…

Finitely-additive measures on the asymptotic foliations of a Markov compactum

An asymptotic expansion is established for time averages of translation flows on flat surfaces. This result, which extends earlier work of A.Zorich and G.Forni, yields limit theorems for translation flows. The argument, close in spirit to that of G.Forni, uses the approximation of ergodic integrals by holonomy-invariant Hoelder cocycles on trajectories of the flows. The space of holonomy-invariant Hoelder cocycles is finite-dimensional, and is given by an explicit construction. First, a symbolic representation for a uniquely ergodic translation flow is obtained following S.Ito and A.M. Vershik…

联合平移谱与曼哈顿流形

The joint translation spectrum and Manhattan manifolds

本文定义了平移锥和联合平移谱，将其与高维曼哈顿流形的梯度函数像等同，并推导出该谱的性质，包括其由随机游走漂移向量闭包给出。

曲线复形上的极小渐近平移长度与映射环面的同调

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

本文研究了曲线复形上极小渐近平移长度的上下界，并构造了非Torelli的pseudo-Anosov映射类以探讨其在亏格趋于无穷时的平移长度衰减限制。

Limit theorems for rank-one Lie groups

We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more precise asymptotic formula that describes their deviations from the limit.

Limit Theorems for Translation Flows

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).

纤维化3-流形的伪Anosov单值群的渐近平移长度与正规生成

Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds

该研究针对双曲纤维化3-流形，以曲线复形上的渐近平移长度为工具，得到伪Anosov单值群平移长度上界、正规生成性质，并证明曲线复形归一化渐近平移长度一般无法连续延拓到纤维化面。

Lie群上Kuramoto--Lohe型同步模型中锁相态的渐近形成与轨道稳定性

Asymptotic formation and orbital stability of phase-locked states in Kuramoto--Lohe type synchronization models on Lie groups

本文提出Lie群上的同步模型，并建立框架分析锁相态的渐近形成与轨道稳定性，作为与Ha和Ko先前工作的延续。

Complexity of Cut-and-Project Sets of Polytopal Type in Special Homogeneous Lie Groups

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups we can establish that the complexity function asymptotically behaves like $r^{homdim(G) dim(H)}$. Further we generalize the concept of acceptance domains to locally compact second countable groups.

Minimal pseudo-Anosov translation lengths on the complex of curves

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a_1/ g^2$ and $a_2/g^2$.

On Sharp Bounds for the Dynamic Asymptotic Dimension

We prove the dynamic asymptotic dimension of a free isometric action on a space of finite doubling dimension is either infinite or equal to the asymptotic dimension of the acting group; and give a full description of the dynamic asymptotic dimension of translation actions on compact Lie groups in terms of the amenability and asymptotic dimension of the acting group.

Lie Bracket Approximation of Extremum Seeking Systems

Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing beh…

在有向度量树上的翻译半群动力学

Dynamics of the translation semigroup on directed metric trees

研究有向度量树上的加权L^p空间中左翻译半群的动力学，推导出强连续性的必要充分条件，并通过树结构中权重的渐近衰减特性刻画超循环性和弱混属性。

Limit theorems for translation flows (published version)

The aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and to give a limit theorem for these flows (Theorem 2).