Charles Holton, Charles Radin, Lorenzo Sadun
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.
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In this article, we extend H. Matui and H. Lin's notions of approximate $K$-conjugacies and $C^*$-strongly approximate conjugacies to general minimal dynamical systems. In particular, upon modifying a result of the existence of minimal skew products, we answer a question of H. Lin and show that, associated with any Cantor minimal system $(K,\tildeα)$, there is a class $R_0(\tildeα)$ of minimal skew products on $K\timesΩ$, such that for any two rigid homeomorphisms $α\in R_0(\tildeα)$ and $β\in R_0(\tildeβ)$, the notions of approximate $K$-conjugacy and $C^*$-strongly approximate conjugacy coin…
We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon, especially with regard to invariant subspaces in the dynamics.
Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity. We describe some of the techniques and recent results that allow differentiable conjugacies to be defined for standard bifurcations, and explain how this leads to a new class of normal forms. Closed-form expressions for differentiable conjugacies exist between some chaotic maps, and we describe some of the constraints that make it possible to recognise when…
To understand an aperiodic tiling (or a quasicrystal modeled on an aperiodic tiling), we construct a space of similar tilings, on which the group of translations acts naturally. This space is then an (abstract) dynamical system. Dynamical properties of the space (such as mixing, or the spectrum of the translation operator) are closely related to bulk properties of the individual tilings (such as the diffraction pattern). The topology of the space of tilings, particularly the Cech cohomology, gives information on how the original tiling can be deformed. Tiling spaces can be constructed as inver…
Let P be an object such as tiling, Delone set and weighted Dirac comb. There corresponds a dynamical system to P, called the corresponding dynamical system. Such dynamical systems are geometric analogues of symbolic dynamics. It is well-known that there are correspondences between geometric properties of P and properties of the corresponding dynamical system. In this article we give a new correspondence. In other words, we characterize the property that the group of topological eigenvalues for the corresponding dynamical system is not discrete, in terms of a geometric property of P.
We investigate the dynamics of substitution subshifts and their associated tiling spaces. For a given subshift, the associated tiling spaces are all homeomorphic, but their dynamical properties may differ. We give criteria for such a tiling space to be weakly mixing, and for the dynamics of two such spaces to be topologically conjugate.
A tiling is a cover of R^d by tiles such as polygons that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in tilings. From a tiling, we can construct a dynamical system which encodes the nature of the tiling. In the literature, properties of this dynamical system were investigated by studying how patches distribute in each tiling. In this article we conversely research distribution of patches from properties of the corresponding dynamical systems. We show periodic structures are hidden in tilings which are not necessarily periodic. Our r…
We investigate substitution subshifts and tiling dynamical systems arising from the substitutions (1) θ: 0 \rightarrow 001,1 \rightarrow 11001 and (2) η: 0 \rightarrow 001,1 \rightarrow 11100. We show that the substitution subshifts arising from θand ηhave minimal self-joinings and are mildly mixing. We also give a criterion for 1-dimensional tiling systems arising from θor ηto have minimal self-joinings. We apply this to obtain examples of mildly mixing 1-dimensional tiling systems.
Overlap coincidence in a self-affine tiling in $\R^d$ is equivalent to pure point dynamical spectrum of the tiling dynamical system. We interpret the overlap coincidence in the setting of substitution Delone set in $\R^d$ and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program. We give the program and apply it to several examples. In the course the proof of the algorithm, we show a variant of the conjecture of Urbański (Solomyak \cite{Solomyak:08}) on the Hausdorff dimension of the boundaries of fractal tiles.
This paper is about the tiling dynamical systems approach to the study of aperiodic order. We compare and contrast four related types of systems: ordinary (one-dimensional) symbolic systems, one-dimensional tiling systems, multidimensional $Z^d$-systems, and multidimensional tiling systems. Aperiodically ordered structures are often hierarchical in nature, and there are a number of different yet related ways to define them. We will focus on what we are calling "supertile construction methods": symbolic substitution in one and many dimensions, S-adic sequences, self-similar and pseudo-self-simi…
After investigating by examples the unusual and striking elementary properties of the Penrose tilings and the Arnold cat map, we associate a finite symbolic dynamics with finite grammar rules to each of them. Instead of studying these Markovian systems with the help of set-topology, which would give only pathological results, a noncommutative approximately finite C*-algebra is associated to both systems. By calculating the K-groups of these algebras it is demonstrated that this noncommutative point of view gives a much more appropriate description of the phase space structure of these systems …
We investigate the role of the proximality relation for tiling dynamical systems. Under two hypotheses, namely that the minimal rank is finite and the set of fiber distal points has full measure we show that the following conditions are equivalent: (i) proximality is topologically closed, (ii) the minimal rank is one, (iii) the continuous eigenfunctions of the translation action span the L^2-functions over the tiling space. We apply our findings to model sets and to Meyer substitution tilings. It turns out that the Meyer property is crucial for our analysis as it allows us to replace proximali…
We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well-studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from t…
We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such systems which cannot be achieved by working within a finite window. By considering 1-dimensional tiling systems, which are the same as flows under functions on subshifts with finite alphabets of symbols, we construct a `simple' code which is not `local', a local code which is not simple, and a continuous code which is neither local nor simple.
We consider self-affine tilings in the Euclidean space and the associated tiling dynamical systems, namely, the translation action on the orbit closure of the given tiling. We investigate the spectral properties of the system. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $ϕ$ for the tiling. Assuming that $ϕ$ is diagonalizable over $\C$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is …
Substitution tilings with transcendental inflation factor
本文针对任意λ>2，在无限字母表上构造替换，生成具有超越膨胀因子的替换平铺，证明相关动力系统严格遍历，并提供显式构造。
We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $σ$-finite invariant measures which are positive and f…
Generalized Integer Partitions, Tilings of Zonotopes and Lattices
本文利用动力系统与序理论，证明广义整数分拆集合构成分配格，并刻画了二维 zonotopes 铺砌集合为分配格的不交并，深化了对翻转铺砌行为的理解。
On the topological dynamical system of tilings
本文提供平铺空间紧致性定理的自包含完整证明，填补文献空白，并详细解释常被省略的论证，以促进该基本结果的理解。
The squiral inflation rule is equivalent to a bijective block substitution rule and leads to an interesting lattice dynamical system under the action of $\mathbb{Z}^{2}$. In particular, its balanced version has purely singular continuous diffraction. The dynamical spectrum is of mixed type, with pure point and singular continuous components. We present a constructive proof that admits a generalisation to bijective block substitutions of trivial height on $\mathbb{Z}^{d}$.
Dynamics and topology of the Hat family of tilings
Hat家族镶嵌的动力学与拓扑结构研究，通过CAP镶嵌分析揭示其自相似性质及纯点动力学谱。
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and compute it for various examples. We also extend our analysis to more general dynamical systems.
In the article "Construction of the continuous hull for the combinatorics of a regular pentagonal tiling of the plane" we constructed a compact topological space for the combinatorics of "A regular pentagonal tiling of the plane", which we call the continuous hull. We also constructed a substitution map on the space which turns out to be a homeomorphism, and so the pair given by the continuous hull and the substitution map yields a dynamical system. In this paper we show how we can write this dynamical system as another dynamical system given by an inverse limit and a right shift map.
We study deviation of ergodic averages for dynamical systems given by self-similar tilings on the plane and in higher dimensions. The main object of our paper is a special family of finitely-additive measures for our systems. An asymptotic formula is given for ergodic integrals in terms of these finitely-additive measures, and, as a corollary, limit theorems are obtained for dynamical systems given by self-similar tilings.
Dynamical Invariants from Asymptotic Composants
本文提出了一种简单算法，用于确定一维原始膨胀镶嵌空间的渐近组成部分，以区分不同的MLD类镶嵌，并展示了其与轨道分离维度结合的区分能力。
On the long-range order of the Spectre tilings
该研究探讨Spectre非周期单调瓦片铺砌的长程有序性，发现其铺砌族的拓扑动力学性质与Hat铺砌相似，相关特性源于第一Čech上同调极小。
A relatively simple substitution for the Robinson tilings is presented, which requires only 56 tiles up to translation. In this substitution, due to Joan M. Taylor, neighboring tiles are substituted by partially overlapping patches of tiles. We show that this overlapping substitution gives rise to a normal primitive substitution as well, implying that the Robinson tilings form a model set and thus have pure point diffraction. This substitution is used to compute the Cech cohomology of the hull of the Robinson tilings via the Anderson-Putnam method, and also the dynamical zeta function of the s…
We introduce a new dynamical system that we call "tiling billiards," where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances with negative indices of refraction. We investigate several special cases where the planar tiling is created by dividing the plane by lines, and we describe the results of computer experiments.
Directional Expansiveness for Rd-Actions and for Penrose Tilings
本文研究 \\(\mathbb{R}^d\\)-作用的方向膨胀性，证明 \\(\mathbb{R}^2\\) 有限局部复杂度镶嵌系统中弱膨胀与强膨胀等价，并确定彭罗斯镶嵌恰有五个非膨胀方向，同时证明其两种王氏瓷砖形变系统与原系统拓扑共轭。
Orbit separation dimension as complexity measure for primitive inflation tilings
本文为本原膨胀平铺的平移动力系统开发了轨道分离维数的计算方法，并利用系统性质显式计算OSD，提供了非平凡示例类。
Matching dynamical systems, through different forms of conjugacies and equivalences, has long been a fundamental concept, and a powerful tool, in the study and classification of nonlinear dynamic behavior (e.g. through normal forms). In this paper we will argue that the use of the Koopman operator and its spectrum is particularly well suited for this endeavor, both in theory, but also especially in view of recent data-driven algorithm developments. We believe, and document through illustrative examples, that this can nontrivially extend the use and applicability of the Koopman spectral theoret…
We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion…
We prove that if a primitive and non-periodic substitution is injective on initial letters, constant on final letters, and has Pisot inflation, then the R-action on the corresponding tiling space has pure discrete spectrum. As a consequence, all beta-substitutions for beta a Pisot simple Parry number have tiling dynamical systems with pure discrete spectrum, as do the Pisot systems arising, for example, from the Jacobi-Perron and Brun continued fraction expansions.
Quantitative weak mixing of self-affine tilings
本文将一维自相似铺砌系统谱测度的 log-Hölder 正则性结果推广到高维自仿射铺砌，并通过谱参数空间的一致估计得到弱混合的对数量化速率。
Symplectic Tiling Billiards on Complete Affine Tori
本文研究环面上非欧几里得几何产生的平面铺层上的辛铺层台球动力系统，通过定义相关仿射结构的开子集，分析其动力学并证明发散轨迹的稳定性结果，引入适配非欧几里得仿射铺层的稀薄性概念。
Given a finite set ${S_1...,S_k}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\rd$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in {1,...,k}^{\mathbb{N}}$. The action by translations on such spaces gives rise to uniquely ergodic dynamical systems. In this paper we investigate the rate of convergence for ergodic limits of patches frequencies and prove that these limits vary continuously with $\bar a$.
Torsors and tilings from toric toggling
本文研究圈图独立集切换的环面类比，用网格编码轨道，通过交换双射构造蛇群与衔尾蛇群的torsors，并刻画圆柱与环面的平行四边形铺砌。
Wind-tree tiling billiards and their trapping strips
研究风树瓷砖弹道系统，证明几乎所有配置中垂直初始方向的轨迹被无限平面带捕获，类比于Eaton透镜中的光传播。
Semigroups, groups, and algebras of dynamical origin
本文综述了与具丰富部分对称性对象相关的动力起源代数结构的最新成果，重点详述收缩自相似逆半群的定义构造，辅以黄金分割旋转半群、Penrose铺砌半群等示例说明。
The paper establishes an equivalence between pure point diffraction and certain types of model sets, called inter model sets, in the context of substitution point sets and substitution tilings. The key ingredients are a new type of coincidence condition in substitution point sets, which we call algebraic coincidence, and the use of a recent characterization of model sets through dynamical systems associated with the point sets or tilings.
There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with a common expansive map $Q$. Our first result gives a partial answer to a question of Lagarias and Wang: we char…
Overlapping substitutions and tilings
本文提出重叠替换的概念，通过非整数矩阵定义拼块频率，并证明其动力系统唯一测度性，同时构造了多种一维和高维例子。
We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses geometric representations of stepped surfaces and related polygonal tilings, as well as self-replicating and periodic tilings made of Rauzy fractals. We apply our theory to an infinite family of substitutions. For this family, we analyze and interpret in a new combinatorial way the codings of a domain exchange defined on the associated fractal domains. We dedu…
We discuss the application of various concepts from the theory of topological dynamical systems to Delone sets and tilings. We consider in particular, the maximal equicontinuous factor of a Delone dynamical system, the proximality relation and the enveloping semigroup of such systems.
In this survey of the spectral properties of substitution dynamical systems we consider primitive aperiodic substitutions and associated dynamical systems: ${\mathbb Z}$-actions and ${\mathbb R}$-actions, the latter viewed as tiling flows. Our focus is on the continuous part of the spectrum. For ${\mathbb Z}$-actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. We give references to complete proofs and emphasize ideas and connections.
In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of $\nats$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets a…
For point sets and tilings that can be constructed with the projection method, one has a good understanding of the correlation structure, and also of the corresponding spectra, both in the dynamical and in the diffraction sense. For systems defined by substitution or inflation rules, the situation is less favourable, in particular beyond the much-studied class of Pisot substitutions. In this contribution, the geometric inflation rule is employed to access the pair correlation measures of self-similar and self-affine inflation tilings and their Fourier transforms by means of exact renormalisati…
For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the regionally proximal relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have a less intricate dynamic and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. …
The theory of substitution sequences and their higher-dimensional analogues is intimately connected with symbolic dynamics. By systematically studying the factors (in the sense of dynamical systems theory) of a substitution dynamical system, one can reach a better understanding of spectral and topological properties. We illustrate this point of view by means of some characteristic examples, including a rather universal substitution in one dimension as well as the squiral and the table tilings of the plane.
We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question of J. C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical spectra.
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