Mélodie Andrieu
We construct an Arnoux-Rauzy word for which the set of all differences of two abelianized factors is equal to $\mathbb{Z}^3$. In particular, the imbalance of this word is infinite - and its Rauzy fractal is unbounded in all directions of the plane.
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We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.
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 that every free group of finite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. This characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and conjugations by free group automorphisms.
The boundary of Rauzy fractal and discrete tilings
本文揭示Rauzy分形点集的分层结构，提出两种基于该结构的构造方法，并以更低计算量构造其边界，同时通过自复制词解释离散平面铺砌。
For substitution satisfying Pisot, irreducible, unimodular condition, a tiling substitution plays a key role in construction of a stepped surface and Rauzy fractal. In this paper we will extend the method to hyperbolic automorphisms on the free group of rank 2 in some class, and obtain set equations of Rauzy fractals by virtue of a tiling substitution. We will also see that the domain exchange transformation on Rauzy fractal is just a two interval exchange transformation.
An elementary proof that the Rauzy gasket is fractal
本文用完全初等的方法证明了 Rauzy 垫片的 Hausdorff 维数严格小于二，从而说明其具有分形性质。
Pisot conjecture and Rauzy fractals
本文证明了遍历理论中关于不可约Pisot替换生成循环序列的Pisot猜想。
In this paper, we study a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular we give explicitly an automaton that generates the boundary of ${\mathcal R}_a$ and using an exotic numeration system we prove that ${\mathcal R}_a$ is homeomorphic to a topological disk.
Rauzy fractals of random substitutions
本文研究了随机置换的拉乌兹分形及其测度，揭示了其与确定性置换分形的联系及连续插值方法。
In this paper, we study arithmetical and topological properties for a class of Rauzy fractals ${\mathcal R}_a$ given by the polynomial $x^3- ax^2+x-1$ where $a \geq 2$ is an integer. In particular, we prove the number of neighbors of ${\mathcal R}_a$ in the periodic tiling is equal to $8$. We also give explicitly an automaton that generates the boundary of ${\mathcal R}_a$. As a consequence, we prove that ${\mathcal R}_2$ is homeomorphic to a topological disk.
We prove that the symbolic dynamical system generated by a purely substitutive Arnoux-Rauzy sequence is measurably conjugate to a toral translation. The proof is based on an explicit construction of a fundamental domain with fractal boundary (a Rauzy fractal) for this toral translation.
We work with attracting subshifts generated by substitutions which are also irreducible parageometric automorphisms of free groups. For such a dynamical system, we construct a tree substitution to approximate the repelling real tree of the automorphism. We produce images of this tree inside the Rauzy fractal when the substitution is irreducible Pisot. We describe the contour of this tree and compute an interval exchange transformation of the circle covering the original substitution.
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 present some topological and arithmetical aspects of a class of Rauzy fractals $\mathcal{R}_{a,b}$ related to the polynomials of the form $P_{a,b}(x)=x^{3}-ax^{2}-bx-1$, where $a$ and $b$ are integers satisfying $-a+1 \leq b \leq -2$. This class has the property that $0$ lies on the boundary of $\mathcal{R}_{a,b}$. We construct explicit finite automata that recognize the boundaries of these fractals, which allows to establish the number of neighbors of $\mathcal{R}_{a,b}$. In particular, we prove that if $2a+3b+4 \leq 0$ then $\mathcal{R}_{a,b}$ is not homeomorphic to a topological disk. We…
We consider the substitution $σ_{a,b}$ defined by $$\begin{array}{rlcl} σ_{a,b}: & 1 & \mapsto & \underbrace{1\ldots 1}_{a}2 \\ & 2 & \mapsto & \underbrace{1\ldots 1}_{b}3 \\ & 3 & \mapsto & 1 \end{array} $$ with $a\geq b\geq 1$. The shift dynamical system induced by $σ_{a,b}$ is measure theoretically isomorphic to an exchange of three domains on a compact tile $\mathcal{T}_{a,b}$ with fractal boundary. We prove that $\mathcal{T}_{a,b}$ is homeomorphic to the closed disk iff $2b-a\leq 3$. This solves a conjecture of Shigeki Akiyama posed in 1997. To this effect, we construct a Hölder continuou…
Let $σ$ be a non-unit Pisot substitution and let $α$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $σ$. In our setting, these fractals are subsets of a certain open subring of the adèle ring $\mathbb{A}_{\mathbb{Q}(α)}$. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, define them in terms of the one-dimensional realization of $σ$ and its dual (in the spiri…
Any infinite sequence of substitutions with the same matrix of the Pisot type defines a symbolic dynamical system which is minimal. We prove that, to any such sequence, we can associate a compact set (Rauzy fractal) by projection of the stepped line associated with an element of the symbolic system on the contracting space of the matrix. We show that this Rauzy fractal depends continuously on the sequence of substitutions, and investigate some of its properties; in some cases, this construction gives a geometric model for the symbolic dynamical system.
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…
Let $β\in(1,2)$ be a Pisot unit and consider the symmetric $β$-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $β$ a $d$-Bonacci number, i.e., Pisot root of $x^d-x^{d-1}-\dots-x-1$ we show that the Rauzy fractals form a multiple tiling with covering degree $d-1$.
The well-known plastic number substitution gives rise to a ternary inflation tiling of the real line whose inflation factor is the smallest Pisot-Vijayaraghavan number. The corresponding dynamical system has pure point spectrum, and the associated control point sets can be described as regular model sets whose windows in two-dimensional internal space are Rauzy fractals with a complicated structure. Here, we calculate the resulting pure point diffraction measure via a Fourier matrix cocycle, which admits a closed formula for the Fourier transform of the Rauzy fractals, via a rapidly converging…
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。
Topological dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain
研究无界域上时滞反应扩散方程全局吸引子的Hausdorff与分形维数估计，通过辅助Hilbert空间重述方程并利用扰动半群方法证明全局解与吸引子存在性，最终获得维数上界。
Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation u_{tt}+αu_t+β(x)u-\Deltau = f(x,u), (t,x)\in[0,+\infty[\timesΩ, u = 0, (t,x)\in[0,+\infty[\times\partialΩin $H^1_0(Ω)\times L^2(Ω) has finite Hausdorff and fractal dimension. Here $Ω$ is a regular, possibly unbounded, domain in $\R^3$ and $f(x,u)$ is a nonlinearity of critical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $f(x,u)$ is…
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