Penrose Tilings、混沌动力系统与代数K-理论

一种新型的菱形彭罗斯铺砌变体

A novel variant of rhombic Penrose tiling

本文提出了一种新型的十fold对称铺砌变体，基于菱形彭罗斯铺砌，通过增加原型瓷砖和双格图案，展现出高密度的十边形图案及更强的超均匀秩序。

Penrose菱形镶嵌的辛几何

The Symplectic Geometry of Penrose Rhombus Tilings

本文的目的是从辛几何的角度来看待Penrose菱形镶嵌。我们发现，每个厚菱形在这样的平铺可以自然地关联到一个高度奇异的4维紧致辛空间，而每个薄菱形可以关联到另一个这样的空间，这两个空间是不变的哈密顿作用下的2维拟，和图像的相应的时刻映射给菱形回来。这两个空间是同构的，但不是辛同构的。

Classical Dimers on Penrose Tilings

We study the classical dimer model on rhombic Penrose tilings, whose edges and vertices may be identified with those of a bipartite graph. We find that Penrose tilings do not admit perfect matchings (defect-free dimer coverings). Instead, their maximum matchings have a monomer density of $81-50φ\approx 0.098$ in the thermodynamic limit, with $φ=\left(1+\sqrt{5}\right)/2$ the golden ratio. Maximum matchings divide the tiling into a fractal of nested closed regions bounded by loops that cannot be crossed by monomers. These loops connect second-nearest neighbour even-valence vertices, each of whi…

Penrose铺砌是一种量子纠错码

The Penrose Tiling is a Quantum Error-Correcting Code

本文揭示非周期性Penrose铺砌可构成一种通过量子几何编码、并可纠正任意有限区域局部错误与擦除的新型量子纠错码，还给出基于Ammann-Beenker和Fibonacci铺砌的多种推广。

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.

准晶：5维晶格在2维和3维中的投影

Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions

我们表明，广义彭罗斯平铺可以通过投影到二维的5维晶格的切割平面，而3-D准周期晶格重叠的单位细胞是它的投影到3D。文中还给出了广义Penrose镶嵌中所有可能顶点类型的出现频率，以及所有可能重叠的三维单胞类型的出现频率。广义彭罗斯平铺被发现是不可转换的风筝和飞镖模式，也不能被描述的Gummelt重叠的十边形。

AdS $_5\times T^{1,1}$的Penrose极限附近的混沌弦

Chaotic strings in a near Penrose limit of AdS$_5\times T^{1,1}$

研究了AdS $_5\times T^{1,1}$在Penrose极限附近经典弦的混沌运动.已知混沌解出现在$R\times T^{1,1}$上，取决于初始条件。这可能是有趣的问是否混乱持续甚至在彭罗斯极限或没有。在本文中，我们表明，在彭罗斯极限的次领先的修正提供了一个不稳定的分界线，使混沌运动产生的后果崩溃Kolmogorov-Arnold-Moser（KAM）环面。我们的分析是基于推导一个简化的系统组成的两个自由度，假设一个缠绕的字符串附件。然后，我们通过计算庞加莱截面，为混沌的存在提供了支持。在比较的AdS $_5\times T^{1,1}$的情况下，我们认为，没有混沌生活在附近的彭罗斯极限的AdS $_5\times$ S $^5$，从经典的父系统的可积性预期。

Fractal spectral triples on Kellendonk's $C^*$-algebra of a substitution tiling

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $θ$-summable, and respects the hierarchy of the substitution system. To elucida…

Isomorphism of Hierarchical Structures

We consider hierarchical structures such as Fibonacci sequences and Penrose tilings, and examine the consequences of different choices for the definition of isomorphism. In particular we discuss the role such a choice plays with regard to matching rules for such structures.

混沌动力系统态密度中的范霍夫奇点

Van Hove singularities in the density of states of a chaotic dynamical system

通过构造周期微分算子序列并计算其态密度，预测混沌系统的统计特性，以斐波那契铺砌规则生成的周期算子为例，推导出混沌系统极限统计的显式公式，并证明临界值附近的聚类等价于算子态密度中的范霍夫奇点。

Fibonacci链经叶化的非对易环面

Noncommutative Torus from Fibonacci Chains via Foliation

本文利用Fibonacci链的指数序列对Fibonacci链进行了分类，并在F链空间上构造了一个近似有限维（AF）的$C^*$ -代数.关于这个AF-代数的K-理论表明了非交换环面和F-链空间之间的联系。一个非交换环面可以看作是环面上叶状的$C^*$ -代数，它被显式嵌入到F-链空间上的AF-代数中。作为其对应，我们利用构造F-链的割过程得到了F-链空间与环面上Kronecker叶状的叶空间之间的关系。