We study infinite covolume discrete subgroups of higher rank semisimple Lie
groups, motivated by understanding basic properties of Anosov subgroups from
various viewpoints (geometric, coarse geometric and dynamical). The class of
Anosov subgroups constitutes a natural generalization of convex cocompact
subgroups of rank one Lie groups to higher rank. Our main goal is to give
several new equivalent characterizations for this important class of discrete
subgroups. Our characterizations capture "rank one behavior" of Anosov
subgroups and are direct generalizations of rank one equivalents to convex
cocompactness. Along the way, we considerably simplify the original definition,
avoiding the geodesic flow. We also show that the Anosov condition can be
relaxed further by requiring only non-uniform unbounded expansion along the
(quasi)geodesics in the group.

本文主要针对高秩半单 Lie 群的无限底积离散子群进行研究，这些研究的动机是为了从几何、粗几何和动力学的各个角度理解 Anosov 子群的基本特性。我们的主要目标是为这个重要的离散子群提供几个新的等效特征描述，这些特征捕捉了 Anosov 子群的 “秩一行为”，并直接推广了秩一等价于凸紧性的情况。在这个过程中，我们大大简化了原始定义，避免了测地线流。我们还表明，Anosov 条件可以通过要求群中（准）测地线的非统一无界膨胀来进一步放宽。

Anosov 子群：动力学和几何特征

Anosov subgroups: Dynamical and geometric characterizations

We propose a new Markov Chain Monte Carlo (MCMC) method for constrained
target distributions. Our method first maps the $D$-dimensional constrained
domain of parameters to the unit ball ${\bf B}_0^D(1)$. Then, it augments the
resulting parameter space to the $D$-dimensional sphere, ${\bf S}^D$. The
boundary of ${\bf B}_0^D(1)$ corresponds to the equator of ${\bf S}^D$. This
change of domains enables us to implicitly handle the original constraints
because while the sampler moves freely on the sphere, it proposes states that
are within the constraints imposed on the original parameter space. To improve
the computational efficiency of our algorithm, we split the Lagrangian dynamics
into several parts such that a part of the dynamics can be handled analytically
by finding the geodesic flow on the sphere. We apply our method to several
examples including truncated Gaussian, Bayesian Lasso, Bayesian bridge
regression, and a copula model for identifying synchrony among multiple
neurons. Our results show that the proposed method can provide a natural and
efficient framework for handling several types of constraints on target
distributions.

该研究论文展示了一种新的用于处理约束目标分布的马尔可夫链蒙特卡罗 (MCMC) 方法，该方法通过映射参数限制域到单位球面完成对参数空间的增广，因此能够在处理约束的同时自由运动于球面上，并且实验证明此方法可以有效的提高计算效率。