Block Maps between Primitive Uniform and Pisot Substitutions
Ville Salo, Ilkka Törmä
本文证明了：对于所有具有相同控制特征值的本原Pisot或一致置换对，存在一个块映射的有限集，使得对应子移位之间的每个块映射都是这个集的元素，直到移位为止.
In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Conjugacy of unimodular pisot substitutions subshifts to domain exchanges
本文证明任意单模Pisot替换子位移可测共轭于欧氏空间中的区域交换，推广了Rauzy与Arnoux-Ito的工作，无需额外组合条件。
We prove the Pisot Conjecture for beta-substitutions: If beta is a Pisot number, the tiling dynamical system associated with the beta-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) every Pisot number is weakly finitary.
Common dynamics of two Pisot substitutions with the same incidence matrix
替换矩阵不足以完全确定相关的动态，即使在最简单的情况下，因为有许多单词具有相同的阿贝尔化。本文研究了具有相同关联矩阵的两个不同Pisot不可约代换$σ_1$和$σ_2$的正则折线的公共点。我们证明，如果0是内点的Rauzy分形相关的$σ_1$这些公共点可以生成一个替代的字母表上的所谓的“平衡块”。
Pisot Substitution Conjecture and Rauzy Fractals
本文利用前缀-后缀自动机导出的有限型子移位的弱混合性质，证明了Pisot猜想的一个版本，并得出替换动力系统具有纯离散谱。
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 consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a ret…
We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and we give a simple necessary and sufficient condition for the symbolic system to have pure discrete spectrum. This condition gives rise to an algorithm based on computation of automata. To see the power of this criterion, we provide families of substitutions that satisfies the Pisot substitution conjecture: 1) $a \mapsto a^kbc$, $b \mapsto c$, $c \mapsto a$, fo…
MLD Relations of Pisot Substitution Tilings
我们考虑1维，么模Pisot替代平铺与三个间隔，并讨论条件下，这种平铺对局部同构（LI），或相互局部可导（MDL）。为此，我们把置换看作是具有三个生成元的基本自由群的同态。然后，如果两个置换被自由群的一个内自同构共轭，则这两个镶嵌是LI，并且两个置换之间的一个共轭外自同构通常可以用来证明这两个镶嵌是MLD。我们提出了几个例子，说明在这种情况下可能发生的不同现象。特别是，我们展示了如何两个替代tilings可以MLD，即使他们的替代矩阵是不相等的，但只有共轭$GL(n,\mathbb{Z})$。我们还说明了如何（在我们的情况下分形）窗口的MLD平铺可以从彼此重建，并讨论如何共轭群自同构影响替代产生的窗口边界。
The geometry of non-unit Pisot substitutions
令$σ$为非单位皮索置换，并令$α$为相关皮索数。已知可以将某些分形瓦片（所谓的\emph{Rauzy fractals}）与$σ$相关联。在我们的设置中，这些分形是阿黛尔环$\mathbb{A}_{\mathbb{Q}(α)}$的某个开子环的子集。给出了几种定义Rauzy分形的方法，并讨论了它们之间的关系。特别是，我们将Rauzy分形视为某些计数系统的自然几何对象，根据$σ$及其对偶的一维实现来定义它们（本着Arnoux和Ito的精神），并将它们视为特定剪辑和项目方案的多组件模型集的对偶。我们还定义了适合于非单位皮索替换的阶梯曲面。我们提供了基本的拓扑和几何性质的Rauzy分形与非单位Pisot替换，证明了一些平铺结果，并提供关系定义的周期点的$σ$，进变换和域交换的subshifts。我们说明了我们的结果，两个和三个字母替换的例子。
A combinatorial approach to products of Pisot substitutions
我们定义了一个通用的算法框架，以证明纯离散谱的替代符号动力系统与一些无限家庭的Pisot替代。我们专注于家庭作为有限产品的三个字母的替代与多维连分数算法的Brun和Jacobi-Perron。我们的工具包括在重新制定的一些组合标准（重合条件），在离散平面生成使用多维（双）替换的属性。我们还推导出一些拓扑和动力学性质的Rauzy分形，基本的符号动力系统，以及一些数论性质的相关Pisot数。
Sellami and Sirvent conjectured that the balanced pair algorithm fails for the following pair of Pisot substitutions: \[ φ_0: \begin{array}{l} a \mapsto abc b \mapsto a c \mapsto ac \end{array} \quad \text{ and } \quad φ_1: \begin{array}{l} a \mapsto cba b \mapsto a c \mapsto ca \end{array}. \] The conjecture stated the balanced pair algorithm fails because there is no initial balanced pair. In the present note we prove this conjecture using a method based on simultaneous coding of the pair of the fixed points of the morphisms.
We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts they define are not topologically mixing. We then show that they satisfy a weaker mixing property using a numeration system arising from a sequence of lengths of inflated words. Moreover, we provide explicit bounds for the corresponding topological entropy in terms of the defining parameters $n$ and $p$.
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.
This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.
Factors of Pisot tiling spaces and the coincidence rank conjecture
本文研究了Pisot置换平铺空间的结构，特别是平移作用不具有纯离散谱的空间的结构。这样的空间总是一个可测的m到1覆盖的平移作用的一个群称为最大等度连续的因素。整数m是替换的重合秩，当且仅当平铺空间上的平移具有纯离散谱时，m等于1。通过考虑介于平铺空间和它的极大等度连续因子之间的因子，建立了具有重合秩2和奇模扩张的一维Pisot置换平铺空间的上同调的一个下界.符合秩猜想，对于符合秩2，是一个推论。
Proximality in Pisot Tiling Spaces
一个置换$\vp$是强Pisot的，如果它的阿贝尔化矩阵是非奇异的，并且除了Perron-Frobenius特征值之外的所有特征值的模都小于1。对于强Pisot $\vp$，满足一个没有循环的条件，并为该平移流的平铺空间$\T_\vp$具有纯离散谱，我们描述了收集$\T^P_\vp$对邻近平铺$\T_\vp$在一个自然的方式作为一个替代平铺空间。我们证明了，如果$ψ$是另一个这样的替代，那么$\T_\vp $和$\T_ψ$同胚当且仅当$\T^P_\vp$和$\T^P_ψ$同胚。我们利用这个不变量来区分平铺空间，其他已知的不变量是无效的。此外，我们表明，强Pisot置换，纯离散谱流的相关的平铺空间是等价的接近是一个封闭的关系上的平铺空间。
We show that, in any dimension greater than one, there are an abelian variety, a smooth rational variety and a Calabi-Yau manifold, with primitive birational automorphisms of first dynamical degree $>1$. We also show that there are smooth complex projective Calabi-Yau manifolds and smooth rational manifolds, of any even dimension, with primitive biregular automorphisms of positive topological entropy.
In this article we study symmetric subsets of Rauzy fractals of unimodular irreducible Pisot substitutions. The symmetry considered is reflection through the origin. Given an unimodular irreducible Pisot substitution, we consider the intersection of its Rauzy fractal with the Rauzy fractal of the reverse substitution. This set is symmetric and it is obtained by the balanced pair algorithm associated with both substitutions.
We prove an extension of the well-known Pisot substitution conjecture to the $S$-adic symbolic setting on two letters. The proof relies on the use of Rauzy fractals and on the fact that strong coincidences hold in this framework.
The Branch Locus for One-Dimensional Pisot Tiling Spaces
如果φ是d次Pisot置换，则平铺空间T_φ i上的膨胀和置换同胚φ i通过几何实现因子化为d维螺线管。在这种实现下，Phi-周期渐近平铺的集合对应于投影到d-环面中的分支轨迹上的有限集。我们证明了，如果两个这样的平铺空间同胚，那么由此产生的分支轨迹是相同的某些仿射映射的环面上的行动。
Geometric realization for substitution tilings
给定一个n维替换，其线性展开式是幺模双曲的，我们利用其一维整数Čech上同调的元素，构造了替换诱导动力学与双曲环体自同构的不变集之间的有限对一的半共轭，称为几何实现.若线性展开满足Pisot族条件，且广义返回向量模的秩等于线性展开的广义度，则几何实现的象是整个环面，并与平铺空间上平移作用到最大等度连续因子上的映上重合.得到了Pisot代换猜想的一个高维推广：若线性展开满足Pisot族条件，且平铺空间的一维上同调的秩等于线性展开的广义度，则平铺空间上的平移作用具有纯离散谱.
Geometrical representation of subshifts for primitive substitutions
本文研究 Perron 特征值为 Pisot 单位的本原替换的子移位，构造了可测共轭的域交换，给出环面平移有限扩张条件，证明弱不可约 Pisot 替换子移位性质，并提供计算伪幺模替换子移位特征值的算法。
Generalized $β$-expansions, substitution tilings, and local finiteness
对于一个相当普遍的类的二维平铺替代，我们证明了，如果长度扩展$β$是一个Pisot数，那么由替代定义的平铺必须是局部有限的。我们还给出了一个矩形瓦片上二维替换的简单示例，具有非Pisot长度扩展$β$，使得替换所允许的瓦片都不是局部有限的。这两个结果的证明是有效的一维，并涉及到某种类型的广义$β$ -变换的想法。
One of the simplest non-Pisot substitution rules is investigated in its geometric version as a tiling with intervals of natural length as prototiles. Via a detailed renormalisation analysis of the pair correlation functions, we show that the diffraction measure cannot comprise any absolutely continuous component. This implies that the diffraction, apart from a trivial Bragg peak at the origin, is purely singular continuous. En route, we derive various geometric and algebraic properties of the underlying Delone dynamical system, which we expect to be relevant in other such systems as well.
Terminal Blocks of Primes in Pisot Numeration Systems
本文在 Pisot 数字系统中证明了固定终端块词的素数定理，解决了 Zeckendorf 猜想，并证明了 Tribonacci 素数定理，同时通过环面模型获得了多项统计性质。
Substitution Tilings and Separated Nets with Similarities to the Integer Lattice
我们表明，任何原始的替代平铺的平面创建一个单独的网络是biLipschitz的整数格。然后证明了如果H是欧几里得空间中的本原Pisot代换，对于每个对应于平铺空间的某个平铺的分离网Y，存在Y与整数格之间的双射F，使Y的每个元素平移一个有界距离.作为一个推论，我们得到，我们有这样一个F的任何分离网，对应于彭罗斯瓷砖。证明依赖于Laczkovich，Burago和Kleiner的结果。
Eigenfunctions for substitution tiling systems
我们证明了，对于唯一遍历${\bf R}^d$行动与一个原始的替代平铺有限的局部复杂性，每个可测量的特征函数符合连续函数几乎处处。因此，拓扑弱混合等价于测度论弱混合。如果置换的扩张映射是由$θ>1$进行的纯扩张，并且置换有一个不动点，则弱混合的失败等价于$θ$是一个Pisot数。
Invariant measures for subshifts arising from substitutions of some primitive components
引入了一些本原分支的置换类。一个双边subshift所产生的替代一些原始组件被分解成成对不相交，局部紧凑，移位不变的集，在每个不变的Radon措施是唯一的缩放。当唯一的不变测度有限时，它可以用关联矩阵的特征值来刻画。
Division Point Measures from Primitive Substitutions
本文推广了Olli关于替换引起的点测度极限的结果。我们考虑$\mathbb{R}^2$中有限多边形集上的一般本原代换，并证明了某些原子测度的极限均收敛于勒贝格测度.
Invariant measures for non-primitive tiling substitutions
我们考虑欧几里得空间中的自仿射镶嵌置换和相应的镶嵌动力系统。众所周知，在原始情况下，动力系统是唯一遍历的。研究了当置换不是本原的，且平铺动力系统是非极小的时的不变测度。我们证明了所有的遍历不变概率测度都是极小分支支持的，但也存在其它自然遍历不变测度，它们是无穷的.在适当的假设下，我们完全刻画了圆柱集上有限的正的$σ$ -有限不变测度。关键一步是在我们的环境中建立非周期性平铺的可识别性。例子包括“整体Sierpienski垫圈和地毯”瓷砖。对于这样的平铺唯一不变的概率测度支持平凡的周期平铺，但有一个完全支持的$σ$ -有限不变测度，这是局部有限的和唯一的缩放。
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…
Rauzy fractals with countable fundamental group
证明了有限秩的自由群都可以作为平面Rauzy分形的基本群来实现，该分形与一个4-幺模三次Pisot代换有关.这刻画了平面Rauzy分形的所有可数基本群。我们给出了一个明确的建设依赖于两个操作的替代：符号分裂和共轭的自由群自同构。
Linearly repetitive Delone sets are rectifiable
本文证明了，对任意整数$d>0$，欧氏$d$-空间$\RR^d$中的每个线性重复Delone集，直到一个双Lipschitz同胚，都等价于整数格$\ZZ^d$.在$\RR^d$中的Delone集$X$来自$\RR^d$的一个本原置换镶嵌的特殊情况下，给出了置换矩阵特征值的一个条件，该条件意味着对于某个正的$λ$，存在从$X$到格$λ\ZZ^d$的有界位移的同胚.这个条件包括原始的Pisot置换镶嵌，但也涉及更广泛的置换镶嵌集。
On word complexity and topological entropy of random substitution subshifts
本文研究随机替换子位移的词复杂性与拓扑熵，不假设相容性，证明原始随机替换零熵等价于确定性替换，并给出常长度情形熵的计算方法及复杂度函数的分类。
Dynamical Green Functions and Discrete Schrödinger Operators with Potentials Generated by Primitive Invertible Substitution
在本文中，我们建立了一个“字典”之间的离散薛定谔算子和某些仿射三次曲面上的全纯动力学，以前的工作由Cantat，Damanik和Gorodetski。第3.2.2和3.2.3段由Serge Cantat编写，作为附录。
Unique ergodicity, and not, for primitive substitutions on compact alphabets containing isolated points
本文研究紧致字母表上原始替换的唯一遍历性，证明即使对自然数一点紧化，原始性也不充分，并给出强幂收敛与拟紧性等价的充要条件。
Aperiodic substitutional systems and their Bratteli diagrams
本文研究由非本原置换引起的非周期置换动力系统。证明了定常有序Bratteli图的Vershik同胚$ϕ$与非周期置换系统同胚的充要条件是$ϕ$对极小分支的约束不与里程表同胚.我们还证明了每一个由具有嵌套性质的替换所生成的非周期替换系统都同胚于一个平稳有序Bratteli图的Vershik映射。证明了每一个非周期置换系统都是可识别的。研究了$m$ -本原置换类及其派生置换类.我们还讨论了有限秩Cantor动力系统的可扩张性的概念。
Fractal tiles induced by tent maps
本文研究由特殊 Pisot 数相关的帐篷映射诱导的分形几何对象 tent-tiles，探讨了其边界 Hausdorff 维数及诱导的格镶嵌，并揭示了其与自由群替换和自同构诱导的 Rauzy 分形之间的联系。
Orbit equivalent substitution dynamical systems and complexity
对任意本原真代换σ，给出了可数多个两两非同构代换动力系统{（X_{ζ_n}，T_{ζ_n}）}_{n=1}^{ \infty }的显式构造，使得它们都是与（X_σ，T_σ）等价的（强）轨道.我们表明，替代动力系统{（X_{ζ_n}，T_{ζ_n}）}的复杂性是本质上不同的，防止他们同构。给定一个本原（不一定是真的）代换τ，我们找到一个具有尽可能少的顶点数的平稳简单真序Bratteli图，使得相应的Bratteli-Vershik系统与（X_τ，T_τ）是轨道等价的.
Pure point diffractive substitution Delone sets have the Meyer property
证明了纯点衍射的本原置换Delone集是Meyer集。这回答了J.C.拉加里亚斯我们还表明，原始的替代Delone集，是一个迈耶集相当于有一个相对密集的布拉格峰。证明是基于平铺动力系统和衍射和动力学谱之间的联系。
We study the topology and dynamics of subshifts and tiling spaces associated to non-primitive substitutions in one dimension. We identify a property of a substitution, which we call tameness, in the presence of which most of the possible pathological behaviours of non-minimal substitutions cannot occur. We find a characterization of tameness, and use this to prove a slightly stronger version of a result of Durand, which says that the subshift of a minimal substitution is topologically conjugate to the subshift of a primitive substitution. We then extend to the non-minimal setting a result obta…
An exceptional set of uniformly spread Kakutani tilings of the line
研究发现五个特定参数下，Kakutani铺砌能形成均匀分布的晶格位移，结合多项式分类与替代规则分析得出这一结论。
On a Family of Random Noble Means Substitutions
本文推广了本原替换规则局部混合的概念，研究由有限个定义同一双侧离散动力壳的本原替换构成的noble means族，给出其随机情形下拓扑熵、遍历性及衍射测度谱型的相关结果。
Symbolic recurrence plot for uniform binary substitutions
本文研究均匀二元替换不动点的符号递归图，证明本原非周期替换的对角线长度集合密度为零但存在长度时其起点密度为正，而非本原替换虽含任意长度对角线但各长度密度均为零。
We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals $\sqrt{q}$ in absolute value. The proof is based on results of M. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.
This paper explores the range of bounded speedups in the topological category. Bounded speedups represent both a strengthening of topological speedups as defined in [A 16] and a generalization of powers of a transformation. Here we show that bounded speedups preserve the structure of two classical minimal Cantor systems. Specifically, a minimal bounded speedup of an odometer is a conjugate odometer, and a minimal bounded speedup of a primitive substitution is again a primitive substitution, though it is never conjugate to the original substitution system. Further, we give bounds on the topolog…
$C^*$-algebras arising from substitutions
本文引入了一个与真本原替换相关联的$C^{\ast}$ -代数.证明了$C^{\ast}$ -代数是单的纯无限代数，它包含相应Cantor极小系的Cuntz-Krieger代数和交叉积$C^{\ast}$ -代数.我们计算了$K$ -群。
We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.
We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.
We prove a general version of the classical Perron-Frobenius convergence property for reducible matrices. We then apply this result to reducible substitutions and use it to produce limit frequencies for factors and hence invariant measures on the associated subshift. The analogous results are well known for primitive substitutions and have found many applications, but for reducible substitutions the tools provided here were so far missing from the theory.
Geometric realizations of two dimensional substitutive tilings
我们定义了二维拓扑替换。欧氏平面或双曲平面的平铺是可替换的，如果底层的2-复形可以通过2维拓扑替换的迭代得到。证明了双曲平面$\mathbb{H}^2$不存在本原代换镶嵌。然而，我们给出了一个替代平铺的$\Hyp^2$，这是非原始的例子。
Random noble means substitutions
研究了一类本原替换规则族在贵族平均膨胀乘子下的随机局部混合问题。这扩展了Godrèche和Luck在1989年引入的随机斐波那契例子。我们讨论了相应的动力系统的结构，并确定了它们的熵、各态历经的不变测度和衍射谱。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。