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)

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

Arithmetic Dynamics

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a "dynamical" sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: (1) Beta-expansions, i.e., the radix expansions in non-integer bases; (2) "Rotational" expansions which arise in the problem of encoding of irrational rotations of the circle; (3) Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodic-theoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create "redundant" representations (those whose space of "digits" is a priori larger than necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep

On univoque Pisot numbers

We study Pisot numbers $\beta \in (1, 2)$ which are univoque, i.e., such that there exists only one representation of 1 as $1 = \sum_{n \geq 1} s_n\beta^{-n}$, with $s_n \in \{0, 1\}$. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.Comment: Accepted by Mathematics of COmputatio

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

Minimal weight expansions in Pisot bases

For applications to cryptography, it is important to represent numbers with a small number of non-zero digits (Hamming weight) or with small absolute sum of digits. The problem of finding representations with minimal weight has been solved for integer bases, e.g. by the non-adjacent form in base~2. In this paper, we consider numeration systems with respect to real bases $\beta$ which are Pisot numbers and prove that the expansions with minimal absolute sum of digits are recognizable by finite automata. When $\beta$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, we determine expansions with minimal number of digits $\pm1$ and give explicitely the finite automata recognizing all these expansions. The average weight is lower than for the non-adjacent form

Purely periodic beta-expansions in the Pisot non-unit case

It is well known that real numbers with a purely periodic decimal expansion are the rationals having, when reduced, a denominator coprime with 10. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit: we characterize real numbers having a purely periodic expansion in such a base; this characterization is given in terms of an explicit set, called generalized Rauzy fractal, which is shown to be a graph-directed self-affine compact subset of non-zero measure which belongs to the direct product of Euclidean and p-adic spaces