arXiv subjects

Reem Yassawi

Publications and source records attributed to Reem Yassawi.

At least 19 recordsLinked to original sources

arXiv2026

From some Pisot numerations to topological groups

A Pisot numeration system $U$ for $\mathbb N$ is a sequence of natural numbers generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number. The purpose of this paper is to introduce the analogue of the group of $p$-adic integers for such numerations when they \emph{preserve zeros}, which is equivalent to the `Condition F' introduced by Frougny and Solomyak for $\beta$-numerations. We show that these topological groups $\mathbb Z_U$ project homomorphically onto a torus. Equipping $\mathbb Z_U$ with the appropriate topology, we also show that if $U$ is unimodular, then $\mathbb Z_U$ is continuously isomorphic to a torus.

math.DS↗

arXiv2026

When is the Ellis semigroup a complete conjugacy invariant?

The Ellis semigroup of a topological dynamical system contains algebraic, topological and dynamical information. It is invariant under conjugacy. Despite this wealth of structure, two non-conjugate dynamical systems can have the same Ellis semigroup. We identify a class of minimal dynamical systems inside which this cannot happen, that is, for which the Ellis semigroup is a complete conjugacy invariant.

math.DS↗

arXiv2025

Cobham's theorem for the Gaussian integers

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $\alpha=-m+i $- and $\beta=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $\alpha$ and $\beta$ are multiplicatively independent Gaussian integers, and at least one of $\alpha$, $\beta$ is not an $n$-th root of an integer, then any $\alpha$- and $\beta$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $\alpha$-automatic for any root of an integer $\alpha$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

math.NT↗

arXiv2024

Obstacles to Topological Factoring of Toeplitz shifts

For every Toeplitz sequence $x$ with period structure $(q_i)_{i\geq 1}$, one can identify a period structure ${\bf p}=(p_i)_{i\geq 0}$ which leads to a Bratteli-Vershik realization of the associated Toeplitz shift; we refer to this period structure as {\it constructive}. Let $(X,\sigma,x)$ and $(Y,\sigma,y)$ be Toeplitz shifts where $x\in X$ and $y\in Y$ are Toeplitz sequences with constructive period structures $(p^n)_{n\geq 1}$ and $(q^n)_{n\geq 1}$, respectively. Using the Bratteli-Vershik realization of factor maps between Toeplitz shifts, we prove that if there exists a topological factoring $ \pi:(X,\sigma)\rightarrow (Y,\sigma)$ with $\pi(x)=y$, then $q\mid p$. In particular, if $\pi$ is conjugacy, then $p=q$. We also prove that Toeplitz sequences are mapped to Toeplitz sequences through topological factorings.

math.DS↗

arXiv2024

Algebraic power series and their automatic complexity modulo prime powers

Christol and, independently, Denef and Lipshitz showed that an algebraic sequence of $p$-adic integers (or integers) is $p$-automatic when reduced modulo $p^\alpha$. Previously, the best known bound on the minimal automaton size for such a sequence was doubly exponential in $\alpha$. Under mild conditions, we improve this to a bound whose dominant factor is $p^{\alpha^3 h d / 3}$, where $h$ and $d$ are the height and degree of the minimal annihilating polynomial modulo $p$. We achieve this bound by showing that all states in the automaton are naturally represented in a new numeration system. This significantly restricts the set of possible states. Since our approach embeds algebraic sequences as diagonals of rational functions, we also obtain bounds more generally for diagonals of multivariate rational functions.

math.NT↗

arXiv2024

A dynamical view of Tijdeman's solution of the chairman assignment problem

In 1980, R. Tijdeman provided an on-line algorithm that generates sequences over a finite alphabet with minimal discrepancy, that is, such that the occurrence of each letter optimally tracks its frequency. In this article, we define discrete dynamical systems generating these sequences. The dynamical systems are defined as exchanges of polytopal pieces, yielding cut and project schemes, and they code tilings of the line whose sets of vertices form model sets. We prove that these sequences of low discrepancy are natural codings of toral translations with respect to polytopal atoms, and that they generate a minimal and uniquely ergodic subshift with purely discrete spectrum. Finally, we show that the factor complexity of these sequences is of polynomial growth order $n^{d-1}$, where $d$ is the cardinality of the alphabet.

math.DS↗

arXiv2024

Mahler equations for Zeckendorf numeration

We define generalised equations of Z-Mahler type, based on the Zeckendorf numeration system. We show that if a sequence over a commutative ring is Z-regular, then it is the sequence of coefficients of a series which is a solution of a Z-Mahler equation. Conversely, if the Z-Mahler equation is isolating, then its solutions define Z-regular sequences. This is a generalisation of results of Becker and Dumas. We provide an example to show that there exist non-isolating Z-Mahler equations whose solutions do not define Z-regular sequences. Our proof yields a new construction of weighted automata that generate classical q-regular sequences.

math.NT↗

arXiv2024

Meyer sets, Pisot numbers, and self-similarity in symbolic dynamical systems

Aperiodic order refers to the mathematical formalisation of quasicrystals. Substitutions and cut and project sets are among their main actors; they also play a key role in the study of dynamical systems, whether they are symbolic, generated by tilings, or point sets. We focus here on the relations between quasicrystals and self-similarity from an arithmetical and dynamical viewpoint, illustrating how efficiently aperiodic order irrigates various domains of mathematics and theoretical computer science, on a journey from Diophantine approximation to computability theory. In particular, we see how Pisot numbers allow the definition of simple model sets, and how they also intervene for scaling factors for invariance by multiplication of Meyer sets. We focus in particular on the characterisation due to Yves Meyer: any Pisot or Salem number is a parameter of dilation that preserves some Meyer set.

math.DS↗

arXiv2023

An elementary proof of Bridy's theorem

Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions.

math.NT↗

arXiv2023

Almost automorphic and bijective factors of substitution shifts

In this article we completely characterise constant length substitution shifts which have an almost automorphic factor, or which have a bijective substitution factor. Our approach is algebraic: we study these dynamical properties in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green's R-relation, and the existence of bijective factors in terms of Green's L-relation. Our results are constructive.

math.DS↗

arXiv2023

Torsion-free $S$-adic shifts and their spectrum

In this work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.

math.DS↗

arXiv2023

Semicocycle discontinuities for substitutions and reverse-reading automata

In this article we define the semigroup associated to a substitution. We use it to construct a minimal automaton which generates a substitution sequence u in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the semicocycle discontinuities of u.

math.DS↗

arXiv2022

Coboundaries and eigenvalues of finitary S-adic systems

An S-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary S-adic dynamical system is one where the directive sequence consists of morphisms selected from a finite set. We study eigenvalues and coboundaries for finitary recognizable S-adic dynamical systems, i.e., those where points can be uniquely desubstituted using the given sequence of morphisms. To do this we identify the notions of straightness and essential words, and use them to define a coboundary, inspired by of Host's formalism, which allows us to express necessary and sufficient conditions that a complex number must satisfy in order to be a continuous or measurable eigenvalue. We then apply our results to finitary directive sequences of substitutions of constant length, and show how to create constant-length $S$-adic shifts with non-trivial coboundaries. We show that in this case all continuous eigenvalues are rational and we give a complete description of the rationals that can be an eigenvalue, indicating how this leads to a Cobham-style result for these systems.

math.DS↗

arXiv2021

Lucas congruences for the Apéry numbers modulo $p^2$

The sequence $A(n)_{n \geq 0}$ of Apéry numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $\mathbb{Z}$-linear combination of multiple zeta values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.

math.NT↗

arXiv2020

The Ellis semigroup of bijective substitutions

For topological dynamical systems $(X,T,σ)$ with abelian group $T$ which admit an equicontinuous factor $π:(X,T,σ)\to (Y,T,δ)$ the Ellis semigroup $E(X)$ is an extension of $Y$ by its subsemigroup $E^{fib}(X)$ of elements which preserve the fibres of $π$. We establish methods to compute $E^{fib}(X)$ and use them to determine the Ellis semigroup of dynamical systems arising from primitive aperiodic bijective substitutions. As an application we show that for these substitution shifts, the virtual automorphism group is isomorphic to the classical automorphism group.

math.DS↗

arXiv2020

Tame or wild Toeplitz shifts

We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli-Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fibres over the maximal equicontinuous factor. The ideas are illustrated using the class of substitution subshifts. A body of elaborate examples shows that the assumptions of our results cannot be relaxed.

math.DS↗

arXiv2020

Recognizability for sequences of morphisms

We investigate different notions of recognizability for a free monoid morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$. Full recognizability occurs when each (aperiodic) point in $\mathcal{B}^\mathbb{Z}$ admits at most one tiling with words $σ(a)$, $a \in \mathcal{A}$. This is stronger than the classical notion of recognizability of a substitution $σ: \mathcal{A}^*\to\mathcal{A}^*$, where the tiling must be compatible with the language of the substitution. We show that if $|\mathcal A|=2$, or if $σ$'s incidence matrix has rank $|\mathcal A|$, or if $σ$ is permutative, then $σ$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé (1992) and Bezuglyi, Kwiatkowski and Medynets (2009), by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$-adic shift it generates, and the measurable Bratteli-Vershik dynamical system that it defines.

math.DS↗

arXiv2020

Automaticity and invariant measures of linear cellular automata

We show that spacetime diagrams of linear cellular automata $Φ: {\mathbb F}_p^{\mathbb Z} \to {\mathbb F}_p^{\mathbb Z}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions which are eventually constant. Each automatic spacetime diagram defines a $(σ, Φ)$-invariant subset of ${\mathbb F}_p^{\mathbb Z}$, where $σ$ is the left shift map, and if the initial condition is not eventually periodic then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(σ, Φ)$-invariant measures on ${\mathbb F}_3^{\mathbb Z}$. Finally, given a linear cellular automaton $Φ$, we construct a nontrivial $(σ, Φ)$-invariant measure on ${\mathbb F}_p^{\mathbb Z}$ for all but finitely many $p$.

cs.FL↗