Shigeki Akiyama

• PhD
• Professor (Full) at University of Tsukuba

We give a simple alternative proof that the monotile introduced by [14] is aperiodic.

We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation $x+y+z=1$ where $\pi x$, $\pi y$, $\pi z$ are three angles of a triangle used in the construction and $x$, $y$, $z$...

We introduce a new generalization of Wythoff Nim using three piles of stones. We show that its P-positions have finite difference properties and produce a partition of positive integers. Further, we give a conjecture that the P-positions approximate a semi-line whose slope is described by algebraic numbers of degree 5.

The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s = (\log \lambda)/(\log \sqrt{2}) , \lambda^{3} = \lambda^{2} + 2 . Although the intersection with a generic line has Hausdorff dimension s-1 , we prove that this does not occur for lines with rational parameters. We further describe the intersectio...

Inspired by \cite{DBZ},we generalize the notion of (geometric) substitution rule to obtain overlapping substitutions. For such a substitution, the substitution matrix may have non-integer entries. We give the meaning of such a matrix by showing that the right Perron--Frobenius eigenvector gives the patch frequency of the resulting tiling. We also s...

In the present work, we exhibit a class of self-descriptive sequences that can be explicitly computed and whose frequencies are known. In particular, as a corollary of our main result, we prove that the sequence introduced in \citeBJM23 has the expected frequencies of occurrences.

We consider the width XT(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_T(\omega )$$\end{document} of a convex n-gon T in the plane along the random direction ω∈R/...

We give an alternative simple proof that the monotile introduced by Smith, Myers, Kaplan and Goodman-Strass is aperiodic.

Let Φn(k)(x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Φn(k)(1) for fixed positive integers n. D. H. Lehmer proved that Φn(k)(1)/Φn(1) is a polynomial of the Euler totient function ϕ(n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Φn(k)(1)/Φn(...

We show that for a Salem number $\beta $ of degree d , there exists a positive constant $c(d)$ where $\beta ^m$ is a Parry number for integers m of natural density $\ge c(d)$ . Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$ .

We show that for a Salem number $\beta$ of degree $d$, there exists a positive constant $c(d)$ that $\beta^m$ is a Parry number for integers $m$ of natural density $\ge c(d)$. Further, we show $c(6)>1/2$.

Let $\Phi_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2\Phi^{(3)}_n(1)$ is divisible by $\phi(n)-2$ and $\Phi^{(2k+1)}_n(1)$ is divisible by $\phi(n)-2k$ for $k\ge 2$.

We extend the key formula which intertwines multiplicative Markoff-Lagrange spectrum and symbolic dynamics. The proof uses complex analysis and elucidates the strategy of the problem. Moreover, the new method applies to a wide variety of polynomials possibly having multiple roots. We derive several consequences of this formula, which are expected o...

We consider the width $X_T(\omega)$ of a convex $n$-gon $T$ in the plane along the random direction $\omega\in\mathbb{R}/2\pi \mathbb{Z}$ and study its deviation rate: $$ \delta(X_T)=\frac{\sqrt{\mathbb{E}(X^2_T)-\mathbb{E}(X_T)^2}}{\mathbb{E}(X_T)}. $$ We prove that the maximum is attained if and only if $T$ degenerates to a $2$-gon. Let $n\geq 2$...

We give asymptotic formulas for the number of balanced words whose slope α and intercept ρ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case, the error term is deduced using an inequality of large sieve type.

We study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that...

We give asymptotic formulas for the number of balanced words whose slope $\alpha$ and intercept $\rho$ lie in a prescribed rectangle. They are related to uniform distribution of Farey fractions and Riemann Hypothesis. In the general case, the error term is deduced using an inequality of large sieve type.

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation. We investigate asymptotic properties of geometric progressions modulo one and observe significantly analogous results on the setL(α)={lim supn→∞‖ξαn‖|ξ∈R}, where ‖x‖ is the distance from x to the nearest integer. First, we show that L(α) is closed in [0,1...

Motivated by phyllotaxis in botany, the angular development of plants widely found in nature, we give a simple mathematical characterization of Delone sets on spirals.

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. This gives another characterization of the Pisot substitution conjecture. For an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and give a simple necessary and sufficient co...

Primitive substitution tilings on {\bb R}^d whose expansion maps are unimodular are considered. It is assumed that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, a cut-and-project scheme can be constructed with a Euclidean internal space. Under some additional condition, it is shown that...

We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitutio...

We study the repetition of patches in self-affine tilings. In particular, we study the existence and non-existence of arithmetic progressions. We first show an arithmetic condition of the expansion map implies the non-existence of one-dimensional arithmetic progressions in self-affine tilings. Next, we show that the existence of full-rank infinite...

We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.

This book presents a panorama of recent developments in the theory of tilings and related dynamical systems. It contains an expanded version of courses given in 2017 at the research school associated with the Jean-Morlet chair program. Tilings have been designed, used and studied for centuries in various contexts. This field grew significantly aft...

Markoff-Lagrange spectrum uncovers exotic topological properties of Diophantine approximation by arithmetic progressions. We observe significantly analogous results on Diophantine approximation by geometric progressions. Letting $\|x\|$ be the distance from $x$ to the nearest integer, we study the set ${\mathcal L}(\alpha)=\ \{\ .\limsup_{n\to \inf...

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$ . We give a sufficient condition for $T$ and $S$ to have the same...

Let $\beta >1$ be an integer or generally a Pisot number. Put $T(x) = \{ \beta x \}$ on $[0,1]$ and let $S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \beta^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.

We show that a constant angle progression on the Fermat spiral forms a Delone set if and only if its angle is badly approximable.

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic t...

We consider Pisot family substitution tilings in Rd whose dynamical spectrum is pure point. There are two cut-and-project schemes (CPSs) which arise naturally: one from the Pisot family property and the other from the pure point spectrum. The first CPS has an internal space Rm for some integer m∈N defined from the Pisot family property, and the sec...

We give a sufficient geometric condition for a subshift to be measurably isomorphic to a domain exchange and to a translation on a torus. And for an irreducible unit Pisot substitution, we introduce a new topology on the discrete line and we give a simple necessary and sufficient condition for the symbolic system to have pure discrete spectrum. Thi...

We consider Pisot family substitution tilings in $\R^d$ whose dynamical spectrum is pure point. There are two cut-and-project schemes(CPS) which arise naturally: one from the Pisot family property and the other from the pure point spectrum respectively. The first CPS has an internal space $\R^m$ for some integer $m \in \N$ defined from the Pisot fa...

A strongly nonperiodic tiling is defined as a tiling that does not admit infinite cyclic symmetry. The purpose of this article is to construct, up to isomorphism, uncountably many strongly nonperiodic hyperbolic tilings with a single vertex configuration by a hyperbolic rhombus tile. We use a tile found by Margulis and Mozes [5], which admits tilin...

Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are...

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu_\beta$ to arbitrary given accuracy whenever $\beta$ is algebraic. In particular, if the G...

We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu_\beta$ to arbitrary given accuracy whenever $\beta$ is algebraic. In particular, if the G...

We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in\mathbb{Z}$ satisfy $|A|\leq B\geq 2$ and $M\in\mathbb{Z}^{2\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\ldots,(B-1)v\}$ for some $v\in\mathbb{Z}^2$ such that $v,M v$ are linearly independent....

We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\in\mathbb{Z}$ satisfy $|A|\leq B\geq 2$ and $M\in\mathbb{Z}^{2\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\mathcal{D}=\{0,v,2v,\ldots,(B-1)v\}$ for some $v\in\mathbb{Z}^2$ such that $v,M v$ are linearly independent....

We complete statement and proof for B. Moss\'e's unilateral recognizability theorem. We also provide an algorithm for deciding the unilateral non-recognizability of a given primitive substitution.

We study the limit shape of successive coronas of a tiling, which models the growth of crystals. We define basic terminologies and discuss the existence and uniqueness of corona limits, and then prove that corona limits are completely characterized by directional speeds. As an application, we give another proof that the corona limit of a periodic t...

Every rational number p/q defines a rational base numeration system in which every integer has a unique finite representation, up to leading zeroes. This work is a contribution to the study of the set of the representations of integers. This prefix-closed subset of the free monoid is naturally represented as a highly non-regular tree. Its nodes are...

A nearly linear recurrence sequence (nlrs) is a complex sequence (a n ) with the property that there exist complex numbers A 0,…, A d−1 such that the sequence \(\big(a_{n+d} + A_{d-1}a_{n+d-1} + \cdots + A_{0}a_{n}\big)_{n=0}^{\infty }\) is bounded. We give an asymptotic Binet-type formula for such sequences. We compare (a n ) with a natural linear...

We study a polyhedron with $n$ vertices of fixed volume having minimum surface area. Completing the proof of Toth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for $n\le 12$, some of them are quite unex...

We study a polyhedron with $n$ vertices of fixed volume having minimum surface area. Completing the proof of Fejes Toth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for $n\le 12$, some of them are quit...

We study the invariant measures of a piecewise expanding map in \(\mathbb {R}^m\) defined by an expanding similitude modulo a lattice. Using the result of Bang (Proc Am Math Soc 2:990–993, 1951) on the plank problem of Tarski, we show that when the similarity ratio is at least \(m+1\), the map has an absolutely continuous invariant measure equivale...

It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in t...

It is known that the M\"obius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form $(e^{2\pi i \alpha \beta^{n}g(\beta)})_{n\in \N}$, for a non-decreasing twice differentiable function $g$ with a mild condition. This follows the result we prove in t...

As an application of the boundary parametrization developed in our previous papers, we propose a new method to deduce information on the connected components of the interior of tiles. This gives a systematic way to study the topology of a certain class of self-affine tiles. An example due to Bandt and Gelbrich is examined to prove the efficiency of...

A nearly linear recurrence sequence (nlrs) is a complex sequence $(a_n)$ with the property that there exist complex numbers $A_0$,$\ldots$, $A_{d-1}$ such that the sequence $\big(a_{n+d}+A_{d-1}a_{n+d-1}+\cdots +A_0a_n\big)_{n=0}^{\infty}$ is bounded. We give an asymptotic Binet-type formula for such sequences. We compare $(a_n)$ with a natural lin...

We define and study the corona limit of a tiling, by investigating the signal propagations on cellular automata (CA) on tilings employing the simple growth CA. In particular, the corona limit of Penrose tilings is the regular decagon.

We study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some...

We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for degree $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper by Akiyama-Lee and elucidates the tight relationship betwe...

We show that strong coincidences of a certain many choices of control points are equivalent to overlap coincidence for the suspension tiling of Pisot substitution. The result is valid for degree $\ge 2$ as well, under certain topological conditions. This result gives a converse of the paper by Akiyama-Lee and elucidates the tight relationship betwe...

We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $\beta$. We prove that it has a unique absolutely continuous invariant probability measure equivalent to Lebesgue measure, provided if $\beta$ is sufficiently large. Restricting to a rotation generated...

Our goal is to present a unified and reasonably complete account of the various conjectures, known as Pisot conjectures, that assert that certain dynamical systems arising from substitutions should have pure discrete dynamical spectrum. We describe the various contexts (symbolic, geometrical, arithmetical) in which substitution dynamical systems ar...

Correlation clustering can he modeled in ihe following way. Let A be a nonempty set, and ∼ be a symmetric binary relation on A. Consider a partition (clustering) P of A. We say that two distinct elements a, b ε A are in conflict, if a∼b, but a and b belong to different classes (clusters) of P, or if a ∼ b, however, these elements belong to the same...

Overlap coincidence is an equivalent criterion to pure discrete spectrum of the dynamics of self-affine tilings in RdRd. In the case of d=1d=1, strong coincidence on mm-letter irreducible substitution has been introduced in Dekking (1978) and Arnoux and Ito (2001) which implies that the system is metrically conjugate to a domain exchange in Rm−1Rm−...

Let v (s) d denote the set of coefficient vectors of contractive polynomials of degree d with 2s non-real zeros. We prove that v (s) d can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate a...

By the algorithm implemented in the paper [2] by Akiyama-Lee, we have examined pure discrete spectrum for some special cases of self-affine tilings.

Let $\alpha$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ \alpha\right] =\mathbb{Z}\left[ \alpha\right]$, if and only if $\alpha$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one...

Let \beta_n>1 be a root of x^n-x-1 for n=4,5,... We will prove that \beta_n is not a Parry number, i.e., the associated beta transformation does not correspond a sofic symbolic system. A generalization is shown in the last section.

In this paper we describe all isometries on the special orthogonal group. As an application we give a form of spectrally multiplicative map on the special orthogonal group.

A complex number alpha is said to satisfy the height reducing property if there is a finite subset F of the ring Z of the rational integers such that Z[alpha]=F[alpha]. This problem of finding F has been considered by several authors, especially in contexts related to self affine tilings, and expansions of real numbers in non-integer bases. We cont...

This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the integers and whose subtrees are all distinct. With every node of that tree is then associated a minimal infinite w...

For a fixed k in (-2,2), the discretized rotation on Z^2 is defined by (x,y)->(y,-[x+ky]). We prove that this dynamics has infinitely many periodic orbits.

We compare the growth of the least common multiple of the numbers u a 1 ,...,u a n and |u a 1 ⋯u a n |, where (u n ) n≥0 is a Lucas sequence and (a n ) n≥0 is some sequence of positive integers.

Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can c...

Let m_1,...,m_s be positive integers. Consider the sequence defined by multinomial coefficients: a_n=\binom{(m_1+m_2+... +m_s)n}{m_1 n, m_2 n,..., m_s n}. Fix a positive integer k\ge 2. We show that there exists a positive integer C(k) such that \frac{\prod_{n=1}^t a_{kn}}{\prod_{n=1}^t a_n} \in \frac 1{C(k)} \Z for all positive integer t, if and o...

We present a variant of Ammann tiles consisting of two similar rectilinear hexagons with edge subdivision, which can tile the plane but only in non-periodic ways. A special matching rule, ghost marking, plays a key role in the proof.

Let α be an algebraic integer and assume that it is expanding, i.e., its all conjugates lie outside the unit circle. We show several results of the form Z[α] = B[α] with a certain finite set B ⊂ Z. This property is called height reducing property, which attracted special interest in the self-affine tilings. Especially we show that if α is quadratic...

An extension of the definition of primitivity of a real nonnegative matrix to ma-trices with univariate polynomial entries is presented. Based on a suitably adapted notion of irreducibility an analogue of the classical characterization of real nonnegative primitive matri-ces by irreducibility and aperiodicity for matrices with univariate polynomial...

We give the sufficient and necessary condition of Browder's convergence theorem for one-parameter nonexpansive semigroups which was proved in [T. Suzuki, Browder's type convergence theorems for one-parameter semigroups of nonexpansive mappings in Banach spaces, Israel J. Math., 157 (2007), 239–257]. We also discuss the perfect kernels of topologica...

Quasicrystals are characterized by the diffraction patterns which consist of pure bright peaks. Substitution tilings are commonly used to obtain geometrical models for quasicrystals. We consider certain substitution tilings and show how to determine a quasicrystalline structure for the substitution tilings computationally. In order to do this, it i...

A standard way to parametrize the boundary of a connected fractal tile T is proposed. The parametrization is Hölder continuous from R/Z to ∂T and fixed points of ∂T have algebraic preimages. A class of planar tiles is studied in detail as sample cases and a relation with the recurrent set method by Dekking is discussed. When the tile T is a topolog...

By the m-spectrum of a real number q>1 we mean the set Y^m(q) of values p(q) where p runs over the height m polynomials with integer coefficients. These sets have been extensively investigated during the last fifty years because of their intimate connections with infinite Bernoulli convolutions, spectral properties of substitutive point sets and ex...

Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section.

We consider a class of planar self-affine tiles $ T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a) $ T = M^{ - 1} \cup _{a \in \mathcal{D}} (T + a) generated by an expanding integral matrix M and a collinear digit set $ \mathcal{D} $ \mathcal{D} as follows: $ M = \left( \begin{gathered} 0 - B \hfill \\ 1 - A \hfill \\ \end{gathered} \right),\mathc...

Overlap coincidence in a self-affine tiling in Rd 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 Rd and find an efficient algorithm to check the pure point dynamical spectrum. This algorithm is easy to implement into a computer program....

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...

In 1994, Martin Gardner stated a set of questions concerning the dissection of a square or an equilateral triangle in three similar parts. Meanwhile, Gardner’s questions have been generalized and some of them are already solved. In the present paper, we solve more of his questions and treat them in a much more general context. Let \({D\subset \math...

We give a description of the fundamental group π(△) of the Sierpiński-gasket △. It turns out that this group is isomorphic to a certain subgroup of an inverse limit formed by the fundamental groups Gn of natural approximations of △. This subgroup, and with it π(△), can be described in terms of sequences of words contained in an inverse limit of sem...

This paper introduces explicit conditions for some natural family of polynomials to define Pisot or Salem numbers, and reviews related topics as well as their references.

For r = (r 1 , , r d) ¾ R d the map r : Z d Z d given by r (a 1 , , a d) = (a 2 , , a d , r 1 a 1 + ¡ ¡ ¡ + r d a d) is called a shift radix system if for each a ¾ Z d there exists an integer k 0 with k r (a) = 0. As shown in the first two parts of this series of papers shift radix systems are intimately related to certain well-known notions of num...

For r = (r(1,)...,r(d)) is an element of R-d the mapping tau(r) : Z(d) -> Z(d) given by tau(r)(a(1),...,a(d))= (a(2),...,a(d), - left perpendicular r(1)a(1) + ... + r(d)a(d) right perpendicular), where left perpendicular.right perpendicular denotes the floor function, is called a shift radix system if for each a is an element of Z(d) there exists a...

Number systems and their associated tilings recently drew attention in relation to symbolic dynamical systems, Diophantine approximation, number theory, automata, quasi-crystal and fractal analysis. In this note, we focus on Pisot number systems and its dual tilings and summarize related results.

We study the connectedness of Pisot dual tilings. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4 which have infinitely many connected components. Also we give a simple necessary and sufficient condition for the connectedness...

This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection between pure periodicity and a compact self-similar repres...

The concept of a canonical number system can be regarded as a natural generalization of decimal representations of rational integers to elements of residue class rings of polynomial rings. Generators of canonical number systems are CNS polynomials which are known in the linear and quadratic cases, but whose complete description is still open. In th...

A word generated by coding of irrational rotation with respect to a general decomposition of the unit interval is shown to have an inverse limit structure directed by substitutions. We also characterize primitive substitutive rotation words, as those having quadratic parameters.

We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for � ∈ { ±1± p 5 2 , ± √ 2, ± √ 3} that all integer sequences (ak)k2Z satisfying 0 ≤ ak 1 + �ak + ak+1 < 1 are periodic.