具有分离覆盖边界有限性条件的殆自守子移位

Finiteness in Polygonal Billiards on Hyperbolic Plane

\textsc{J. Hadamard} studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating "Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in "rational polygons" on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just 'Subshifts of Finite Type' or their dense subsets. We further show that 'Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate …

Local finiteness and automorphism groups of low complexity subshifts

We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac{\log (c_n(X)/n)}{\log \log \log n} = 0$, then the quotient group $\textrm{Aut}(X,σ) / \langle σ\rangle$ of the automorphism group of $X$ by the subgroup generated by the shift $σ$ is locally finite. We prove that significantly weaker upper bounds on $c_n(X)$ imply the same conclusion if the Gap Conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of $X$ of range $n$ in terms of word complexity, which may be of independent in…

周期点与子移位自同构群的余有限性

Periodic points and residual finiteness of automorphism groups of subshifts

研究子移位的周期点密度与其自同构群余有限性之间的关系，证明在周期点不密集时，自同构群非余有限，并构造了一个非余有限的强不可约子移位例子。

On dynamical finiteness properties of algebraic group shifts

Let $G$ be a group and let $V$ be an algebraic group over an algebraically closed field. We introduce algebraic group subshifts $Σ\subset V^G$ which generalize both the class of algebraic sofic subshifts of $V^G$ and the class of closed group subshifts over finite group alphabets. When $G$ is a polycyclic-by-finite group, we show that $V^G$ satisfies the descending chain condition and that the notion of algebraic group subshifts, the notion of algebraic group sofic subshifts, and that of algebraic group subshifts of finite type are all equivalent. Thus, we obtain extensions of well-known resul…

几乎自同构系统具有不变测度

Almost automorphic systems have invariant measures

本文证明非amenable可数群在紧致空间上的几乎自同构作用（具有Cantor集极大等度连续因子）承认不变概率测度，从而Toeplitz子移位不能检验amenability，并给出相关扩展与稠密性结果。

局部有限-by-虚拟 $\mathbb{Z}$ 群的非自由几乎有限作用

Non-free almost finite actions for locally finite-by-virtually $\mathbb{Z}$ groups

本文研究局部有限-by-虚拟Z群在Cantor集上的非自由极小作用，证明在特定条件下测度几乎有限等价于本质自由，并推出虚拟Z群拓扑自由作用几乎有限，首次仅需拓扑自由性，并给出交叉积C*-代数的新结果及具体例子。