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.
This paper studies tilings related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic beta-expansion. Special focus is given to some quadratic examples.
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.
For a (non-unit) Pisot number $β$, several collections of tiles are associated with $β$-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the $β$-transformation and a Euclidean one made of integral beta-tiles. We show that all these collections (except possibly the periodic translation of the central tile) are tilings if one of them is a tiling or, equivalently, the weak finiteness property (W) holds. We also obtain new results on rational numbers with purely periodic $β$-expansions; in particular, we calculate …
We study rational numbers with purely periodic Rényi $β$-expansions. For bases $β$ satisfying $β^2=aβ+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $γ(β)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $β$-expansion. A simple algorithm for determining the value of $γ(β)$ for all quadratic Pisot numbers $β$ is described.
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 $β$ and the greedy $β$-transformation. In this paper, we consider different transformations generating expansions in base $β$, 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 $β$-expansions. Remarkably, the symmetric $β$-transformation does …
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.
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 $d-1$.
We study real numbers $β$ with the curious property that the $β$-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 $γ(β)$ denote the supremum of the real numbers $c$ in $(0,1)$ such that all positive rational numbers less than $c$ have a purely…
The aim of this paper is to study the connection between different properties related to $β$-expansions. In particular, the relation between two conditions, both ensuring pure discrete spectrum of the odometer, is analysed. The first one is the so-called Hypothesis B for the $G$-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot $β$-numerations.
The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $β$ having the negative finiteness property, that is the set of finite $(-β)$-expansions is equal to $\mathbb{Z}[β^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-β)$-integers. We also give conditions excluding the negative finiteness property.
Let $β>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1,...,m-1\}$ for all $k$ (a $β$-expansion of $x$). It is known that a.e. $x\in I_β$ has a continuum of distinct $β$-expansions. In this paper we prove that if $β$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $β$. When $β<\frac{1+\sqrt5}2$, we show that the set of $β$-expansions grows …
We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $β$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $β>B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $β>B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $ζ$ with all parameters in $\mathbb{Q}(ζ,β)$, it gives a sofic system when $\cos(2π/q) \in \mathbb{Q}(β)$ and $β$ is a Pisot num…
We give a fractal-geometric condition for a measure on [0,1] to be supported on points x that are normal in base n, i.e. such that the sequence x,nx,n^2 x,... equidistributes modulo 1. This condition is robust under C^1 coordinate changes, and it applies also when n is a Pisot number and equidistribution is understood with respect to the beta-map and Parry measure. As applications we obtain new results (and strengthen old ones) about the prevalence of normal numbers in fractal sets, and new results on measure rigidity, specifically completing Host's theorem to multiplicatively independent inte…
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 …
Given an integer $N\ge 2$ and a real number $β>1$, let $Γ_{β,N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{β^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a $β$-expansion of $x$. Let $\mathbf{U}_{β,N}$ be the set of all $x$'s in $Γ_{β,N}$ which have unique $β$-expansions. We give explicit formula of the Hausdorff dimension of $\mathbf{U}_{β,N}$ for $β$ in any admissible interval $[{β}_L,{β}_U]$, where ${β_L}$ is a purely Parry number while ${β_U}$ is a transcendental number whose quasi-greedy expansion of $1$ is related to the classical Thue-…
Let $β\in(1,2)$. Each $x\in[0,\frac{1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1\}$ for all $k$ (a $β$-expansion of $x$). If $β>\frac{1+\sqrt5}{2}$, then, as is well known, there always exist $x\in(0,\frac1{β-1})$ which have a unique $\be$-expansion. In the present paper we study (purely) periodic unique $β$-expansions and show that for each $n\ge2$ there exists $β_n\in[\frac{1+\sqrt5}{2},2)$ such that there are no unique periodic $β$-expansions of smallest period $n$ for $β\leβ_n$ and at least one such expansion for $β>β_n$. Furthermore, we …
