Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection of fractal tiles with shift radix systems. We show that for certain classes of parameters $\mathbf{r}$ these tiles coincide with affine copies of the well-known tiles associated with beta-expansions and canonical number systems. On the other hand, these tiles provide natural families of tiles for beta-expansions with (non-unit) Pisot numbers as well as canonical number systems with (non-monic) expanding polynomials. We also prove basic properties for tiles associated with shift radix systems. Indeed, we prove that under some algebraic conditions on the parameter $\mathbf{r}$ of the shift radix system, these tiles provide multiple tilings and even tilings of the $d$-dimensional real vector space. These tilings turn out to have a more complicated structure than the tilings arising from the known number systems mentioned above. Such a tiling may consist of tiles having infinitely many different shapes. Moreover, the tiles need not be self-affine (or graph directed self-affine)

The geometry of non-unit Pisot substitutions

Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emph{Rauzy fractals}, with $\sigma$. In our setting, these fractals are subsets of a certain open subring of the ad\`ele ring $\mathbb{A}_{\mathbb{Q}(\alpha)}$. 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 $\sigma$ and its dual (in the spirit of Arnoux and Ito), and view them as the dual of multi-component model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of Rauzy fractals associated with non-unit Pisot substitutions, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of $\sigma$, to adic transformations, and a domain exchange. We illustrate our results by examples on two and three letter substitutions.Comment: 29 page

Shift Radix Systems - A Survey

Let $d\ge 1$ be an integer and ${\bf r}=(r_0,\dots,r_{d-1}) \in \mathbf{R}^d$. The {\em shift radix system} $\tau_\mathbf{r}: \mathbb{Z}^d \to \mathbb{Z}^d$ is defined by $$\tau_{{\bf r}}({\bf z})=(z_1,\dots,z_{d-1},-\lfloor {\bf r} {\bf z}\rfloor)^t \qquad ({\bf z}=(z_0,\dots,z_{d-1})^t).$$ $\tau_\mathbf{r}$ has the {\em finiteness property} if each ${\bf z} \in \mathbb{Z}^d$ is eventually mapped to ${\bf 0}$ under iterations of $\tau_\mathbf{r}$. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of $\tau_\mathbf{r}$ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure

Rational self-affine tiles

An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of $\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\mathbf{A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p}$ of the ad\'ele ring $\mathbb{A}_K$, where the factors of the (finite) product are certain $\mathfrak{p}$-adic completions of a number field $K$ that is defined in terms of the characteristic polynomial of $\mathbf{A}$. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with $\mathbb{R}^n \times \prod_\mathfrak{p} \{0\} \simeq \mathbb{R}^n$. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems

Number representation using generalized (−β)(-\beta)-transformation

We study non-standard number systems with negative base $-\beta$. Instead of the Ito-Sadahiro definition, based on the transformation $T_{-\beta}$ of the interval $\big[-\frac{\beta}{\beta+1},\frac{1}{\beta+1}\big)$ into itself, we suggest a generalization using an interval $[l,l+1)$ with $l\in(-1,0]$. Such generalization may eliminate certain disadvantages of the Ito-Sadahiro system. We focus on the description of admissible digit strings and their periodicity.Comment: 22 page

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$-transformation. In this paper, we consider different transformations generating expansions in base $\beta$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$-expansions. Remarkably, the symmetric $\beta$-transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$-transformation. Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits

Rational numbers with purely periodic β\beta-expansion

We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due to Akiyama to characterize algebraic numbers of degree 3 that enjoy this property. This extends results previously obtained in the case of degree 2 by Schmidt, Hama and Imahashi. Let $\gamma(\beta)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely periodic $\beta$-expansion. We prove that $\gamma(\beta)$ is irrational for a class of cubic Pisot units that contains the smallest Pisot number $\eta$. This result is motivated by the observation of Akiyama and Scheicher that $\gamma(\eta)=0.666 666 666 086 ...$ is surprisingly close to 2/3

Algorithm for determining pure pointedness of self-affine tilings

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 Urba\'nski (Solomyak \cite{Solomyak:08}) on the Hausdorff dimension of the boundaries of fractal tiles.Comment: 21 pages, 3 figure

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems