Model sets with Euclidean internal space

膨胀集的分布：从欧几里得几何到双曲几何的旅程

The distribution of dilating sets: a journey from Euclidean to hyperbolic geometry

本文综述并推广常截面曲率流形上渐进膨胀集经覆盖映射投影后的分布结果，并将欧氏情形推广到紧双曲曲面的单位切丛，建立齐次曲线膨胀平移平均的精确渐近展开。

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. In this paper we characterise, by introducing novel variants of shadowing, maps for which every element of $ICT_f$ is equal to (resp. may be approximated by) the $α$-limit set and the $ω$-limit set of the same full trajectory. We construct examples highlighting the difference between these properties.

欧几里得空间之外的纯点衍射与熵

Pure point diffraction and entropy beyond the Euclidean space

本文将纯点衍射 Delone 集的零熵性质推广到 σ-紧局部紧 Abel 群，证明相关动力系统拓扑熵为零，并构造反例说明补片计数熵可能非零、不可沿极限计算甚至无限。

自相似集上的奇异积分与Heisenberg群中Lipschitz调和函数的可移性

Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups

本文研究度量群的小子集（即测度为零且小于满维的子集）上的奇异积分。我们想到的群的主要例子是欧几里得空间和海森堡群。除了获得结果在一个非常一般的设置，这项工作的目的是双重的，我们将延长一些结果在欧几里得空间更一般的内核比以前考虑，我们将获得在海森堡群的一些应用调和（在海森堡意义上）功能的一些结果较早在欧几里得空间。

Shadowing, internal chain transitivity and $α$-limit sets

Let $f \colon X \to X$ be a continuous map on a compact metric space $X$ and let $α_f$, $ω_f$ and $ICT_f$ denote the set of $α$-limit sets, $ω$-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map $f$ has shadowing then every element of $ICT_f$ can be approximated (to any prescribed accuracy) by both the $α$-limit set and the $ω$-limit set of a full-trajectory. Furthermore, if $f$ is additionally c-expansive then every element of $ICT_f$ is equal to both the $α$-limit set and the $ω$-limit set of a full-trajectory. In particular this means that …

定义在 Markov-Cayley 树上的树移位的极限集、内部链传递性与轨道跟踪性

Limit sets, internal chain transitivity and orbital shadowing of tree-shifts defined on Markov-Cayley trees

本文研究了 Markov-Cayley 树上树移位的 $ω$-极限集与伪轨道，通过引入基于完全前缀集的修正定义及投影伪轨道概念，推广了 $d$-树上的相关结论并探究了其跟踪性质。

几乎周期离散集

Almost periodic discrete sets

利用序列空间中的一种特殊度量，给出了$k$维欧几里得空间中概周期集的几何描述。我们证明了概周期集空间的完备性和概周期性的Bochner准则的一些类似性。此外，我们展示了这些集合和概周期测度之间的联系。

Diffusion on Delone sets

We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.