Shift Radix Systems Research Articles

Representations for complex numbers with integer digits

We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

Rational self-affine tiles

An integral self-affine tile is the solution of a set equation A T = ⋃ d ∈ D ( T + d ) \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d) , where A \mathbf {A} is an n × n n \times n integer matrix and D \mathcal {D} is a finite subset of Z n \mathbb {Z}^n . In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Q n × n \mathbf {A} \in \mathbb {Q}^{n \times n} . We define rational self-affine tiles as compact subsets of the open subring R n × ∏ p K p \mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p} of the adèle ring A K \mathbb {A}_K , where the factors of the (finite) product are certain p \mathfrak {p} -adic completions of a number field K K that is defined in terms of the characteristic polynomial of A \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 tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with R n × ∏ p { 0 } ≃ R n \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 the digit set D \mathcal {D} , intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.

Characterization algorithms for shift radix systems with finiteness property

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

Contractivity of three-dimensional shift radix systems with finiteness property

Let d ≥ 1 be an integer and . We define the shift radix system τ r : ℤ d → ℤ d by The shift radix system τ r has the finiteness property if each a ∈ ℤ d is eventually mapped to 0 under iterations of τ r . The mapping τ r can be written as , where R(r) is a d × d matrix and v is a correction term. It has been conjectured that the fact that τ r has the finiteness property implies that all eigenvalues of R(r) are strictly smaller than one in modulus. The aim of the present paper is to prove this conjecture for the case d = 3.

Fractal tiles associated with shift radix systems

Shift radix systems form a collection of dynamical systems depending on a parameter 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 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 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).

$\epsilon$-shift radix systems and radix representations with shifted digit sets

$\epsilon$-shift radix systems and radix representations with shifted digit sets

Generalized radix representations and dynamical systems. IV

For r = ( r 1,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1,…, a d ) = ( a 2, …, a d ,−⌊ r 1 a 1+…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

ON A FAMILY OF THREE TERM NONLINEAR INTEGER RECURRENCES

In the present paper we study sequences defined by the recurrence relation [Formula: see text] for n ≥ 0, where [Formula: see text] the golden ratio. These sequences are related to shift radix systems as well as to β-expansions with respect to Salem numbers.

Three-dimensional symmetric shift radix systems

Shift radix systems have been introduced by Akiyama et al. as a common generalization of β-expansions and canonical number systems. In the present paper we study a variant of them, so-called symmetric shift radix systems which were introduced recently by Akiyama and Scheicher. In particular, for d ∈ N and r ∈ R let (a = (a1, . . . , ad)) τr : Z d → Z, a 7→ (

Remarks on a conjecture on certain integer sequences

The periodicity of sequences of integers ]]> ]]> ]]> ]]> ]]> ]]> ]]> (a_{n})_{n\in\mathbb Z}$ satisfying the inequalities ]]> 0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 (n \in {\mathbb Z}) $$ is studied for real $ \lambda $ with $|\lambda|< 2$. Periodicity is proved in case $ \lambda $ is the golden ratio; for other values of $ \lambda $ statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.

Dynamical directions in numeration

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β-numeration and its generalisations, abstract numeration systems and shift radix systems, as well as G-scales and odometers. A section of applications ends the paper.