Dynamical directions in numeration

DOI10.5802/aif.2233zbMath1138.37005OpenAlexW2509973786MaRDI QIDQ2372766

Guy Barat, Valérie Berthé, Pierre Liardet, Jörg M. Thuswaldner

Publication date: 1 August 2007

Published in: Annales de l'Institut Fourier (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=AIF_2006__56_7_1987_0

zbMATH Keywords

bibliographysymbolic dynamicstilings\(f\)-expansionsRauzy fractalsadditive functionssubshiftsodometersnumerationsum-of-digits functionfibred systems\(\beta\)-numerationsubstitutive dynamical systemsabstract number systemscanonical numeration systemsnumeration scalesshift radix systems

Mathematics Subject Classification ID

Related Items (33)

Symmetric and congruent Rauzy fractals ⋮ On the reducible quintic complete base polynomials ⋮ On the characterization of expansion maps for self-affine tilings ⋮ On single-matrix graph-directed iterated function systems ⋮ Around the Danzer problem and the construction of dense forests ⋮ Digit systems over commutative rings ⋮ On linear combinations of units with bounded coefficients and double-base digit expansions ⋮ Non-normal numbers with respect to Markov partitions ⋮ Fractal tiles associated with shift radix systems ⋮ Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated ⋮ Path sets in one-sided symbolic dynamics ⋮ Numeration and discrete dynamical systems ⋮ Rational matrix digit systems ⋮ Self-affine manifolds ⋮ Borel complexity of sets of normal numbers via generic points in subshifts with specification ⋮ Recurrence in \(\beta \)-expansion over formal Laurent series ⋮ Lochs-type theorems beyond positive entropy ⋮ General Framework ⋮ Normal Numbers and Symbolic Dynamics ⋮ Distribution of the sum-of-digits function of random integers: a survey ⋮ The Thue-Morse and Rudin-Shapiro sequences at primes in principal number fields ⋮ Analysis of a class of spectral pair conditions ⋮ S-adic Sequences: A Bridge Between Dynamics, Arithmetic, and Geometry ⋮ Purely periodic \(\beta \)-expansions in the Pisot non-unit case ⋮ The structure of abelian groups supporting a number system (extended abstract) ⋮ On self-affine tiles whose boundary is a sphere ⋮ Combinatorial and probabilistic properties of systems of numeration ⋮ Noncanonical number systems in the integers ⋮ Substitutive number systems ⋮ Pseudorandomness of the Ostrowski sum-of-digits function ⋮ Number systems and tilings over Laurent series ⋮ On the dichotomy of Perron numbers and beta-conjugates ⋮ Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

Cites Work

• BKM: a new hardware algorithm for complex elementary functions
• Coincidence for substitutions of Pisot type
• Automatic Sequences
• Reprsentation des systmes dynamiques substitutifs non unimodulaires
• Random \beta-expansions
• Two-symbol Pisot substitutions have pure discrete spectrum
• Geodesic laminations as geometric realizations of Pisot substitutions
• Propriétés topologiques et combinatoires des échelles de numération
• The Hausdorff Dimension of the Boundary of a Self-Similar Tile
• Frontière du fractal de Rauzy et système de numération complexe
• Directions in Mathematical Quasicrystals
• On purely periodic beta-expansions of Pisot numbers
• Dynamiques associees a une echelle de numeration
• A theorem of Cobham for non-primitive substitutions
• Canonical number systems, counting automata and fractals
• Generalized quasiperiodic Rauzy tilings
• An RNS Montgomery modular multiplication algorithm
• ORDERED BRATTELI DIAGRAMS, DIMENSION GROUPS AND TOPOLOGICAL DYNAMICS
• Natural extensions for the Rosen fractions
• Substitution dynamical systems : algebraic characterization of eigenvalues
• Canonical number systems in algebraic number fields
• The Construction of Decimals Normal in the Scale of Ten
• On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces
• Good approximation and characterization of subgroups of R = Z
• Pure Discrete Spectrum for One-dimensional Substitution Systems of Pisot Type
• New criteria for canonical number systems
• Répartition des suites (nα)_n∈Net substitutions
• Odometers and systems of numeration
• An Introduction to Symbolic Dynamics and Coding
• On the beta expansion for Salem numbers of degree 6
• Integral Self-Affine Tiles in ℝ ⁿ I. Standard and Nonstandard Digit Sets
• Entropy quotients and correct digits in number-theoretic expansions
• Numeration systems as dynamical systems–introduction
• Three-dimensional symmetric shift radix systems
• Handbook of Elliptic and Hyperelliptic Curve Cryptography
• A ``Binary System for Complex Numbers
• New redundant representations of complex numbers and vectors
• Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains
• Generalized radix representations and dynamical systems II
• Progress in Cryptology – Mycrypt 2005
• Selected Areas in Cryptography
• Mathematical Foundations of Computer Science 2005
• Representations for real numbers
• On a conjecture of Erdös and Szüsz related to uniform distribution mod 1
• An Explicit Expression for Binary Digital Sums
• On the base-dependence of sets of numbers recognizable by finite automata
• Sur les fonctions q-additives ou q-multiplicatives
• The Sum-of-Digits Function of Squares
• An Asymptotic Formula for the Average Sum of the Digits of Integers
• A SIGNED BINARY MULTIPLICATION TECHNIQUE
• A generalization of continued fractions
• Representations for real numbers
• Dynamical systems of algebraic origin
• On a problem in additive number theory
• Numeration systems on a regular language
• On periodic \(\beta\)-expansions of Pisot numbers and Rauzy fractals
• An almost-sure estimate for the mean of generalized \(Q\)-multiplicative functions of modulus 1
• Topological properties of two-dimensional number systems
• Higher dimensional extensions of substitutions and their dual maps
• The Zeckendorf expansion of polynomial sequences
• Geometric study of the beta-integers for a Perron number and mathematical quasicrystals
• Symmetry groups for beta-lattices
• Generalized number systems in Euclidean spaces
• Number expansions in lattices
• Real numbers having ultimately periodic representations in abstract numeration systems
• Substitutions and \(\beta\) systems of numeration
• Integral self-affine tiles in \(\mathbb{R}^n\). II: Lattice tilings
• Gaussian asymptotic properties of the sum-of-digits function
• Sequences, discrepancies and applications
• Substitutions in dynamics, arithmetics and combinatorics
• Transformations on [0,1 with infinite invariant measures]
• On Rauzy fractal
• Numeration systems, fractals and stochastic processes
• On the number of optimal base 2 representations of integers
• Remarks on a conjecture on certain integer sequences
• Odometers on regular languages
• Unimodular Pisot substitutions and their associated tiles
• How to write integers in a non-integral basis
• On the fractal curves induced from the complex radix expansion
• Some remarks about Fibonacci multiplication
• \(\beta\)-expansions in algebraic function fields over finite fields
• Fourier-Bohr spectrum of sequences related to continued fractions
• Power sums of digital sums
• Une nouvelle propriété des suites de Rudin-Shapiro. (A new property of Rudin-Shapiro sequences)
• Some skew product transformations associated with continued fractions and their invariant measures
• Fibonacci multiplication
• Substitution dynamical systems - spectral analysis
• Approximations of real numbers by the sequence \(\{\) n\(\alpha\) \(\}\) and their metrical theory
• Random f-expansions
• Caractérisation des ensembles normaux substitutifs. (Characterization of normal substitutional sets)
• Systèmes de numération et fonctions fractales relatifs aux substitutions. (Numeration systems and fractal functions related to substitutions)
• Almost topological dynamical systems
• The action of diffeomorphism of the circle on the Lebesgue measure
• A construction of transversal flows for maximal Markov automorphisms
• Radix representations of quadratic fields
• Train tracks and automorphisms of free groups
• Power of words and recognizability of fixpoints of a substitution
• Sur la fonction sommatoire de la fonction 'somme des chiffres'
• Point spectra of non-singular flows
• On recognizable sets of integers
• Arithmetic and dynamical properties of the Rauzy fractal
• Hausdorff dimension of boundaries of self-affine tiles in \(\mathbb{R}^N\)
• The natural extension of the \(\beta\)-transformation
• Codification of Pisot endomorphisms of the torus \([0,1[^r\) and simultaneously invariant measures for two homomorphisms of the torus
• Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions
• Modified Jacobi-Perron algorithm and generating Markov partitions for special hyperbolic toral automorphisms
• Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts
• Logic and \(p\)-recognizable sets of integers
• Directed graphs and substitutions
• Symbolic dynamics. One-sided, two-sided and countable state Markov shifts
• A generalization of Cobham's theorem
• Carry propagation in signed digit representations
• On the boundary of self-affine tilings generated by Pisot numbers
• Cubic CNS polynomials, notes on a conjecture of W. J. Gilbert
• Digit systems in polynomial rings over finite fields
• Substitution Delone sets
• Consequences of pure point diffraction spectra for multiset substitution systems
• Additive functions with respect to numeration systems on regular languages
• Functional limit theorems for digital expansions
• Connectedness of geometric representation of substitutions of Pisot type
• On the representation of real numbers using regular languages
• On dynamics related to a class of numeration systems
• Prefix-suffix automaton associated with a primitive substitution
• Combinatorial, ergodic and arithmetic properties of the Tribonacci substitution
• Invariant sets and measures of nonexpansive group automorphisms
• \(\beta\)-expansions and symbolic dynamics
• On symmetric radix representation of Gaussian integers
• Signed bits and fast exponentiation
• A certain finiteness property of Pisot number systems
• The common dynamics of the Tribonacci substitutions
• Self-affine tiling via substitution dynamical systems and Rauzy fractals.
• Tribonacci multiplication
• Bijective and general arithmetic codings for Pisot toral automorphisms
• On simultaneous approximation to \((\alpha,\alpha^2)\) with \(\alpha^3+k\alpha-1=0\).
• On the connectedness of self-affine attractors
• On the characterization of canonical number systems
• Pure discrete spectrum dynamical system and periodic tiling associated with a substitution
• Distribution of the values of \(q\)-additive functions on polynomial sequences
• Self-affine tiles in \(\mathbb{R}^n\)
• Algebraic coding of expansive group automorphisms and two-sided beta-shifts
• Bijective arithmetic codings of hyperbolic automorphisms of the \(2\)-torus, and binary quadratic forms
• A theorem on the Markov periodic approximation in ergodic theory
• Fractal representation of the attractive lamination of an automorphism of the free group.
• Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \(\beta\)-shifts
• Atomic surfaces, tilings and coincidences. II: Reducible case
• Substitutions, abstract number systems and the space filling property
• Diffraction spectra of weighted Delone sets on beta-lattices with beta a quadratic unitary Pisot number
• On the fundamental group of self-affine plane tiles
• On gaps in Rényi \(\beta\)-expansions of unity for \(\beta>1\) an algebraic number
• The topological structure of fractal tilings generated by quadratic number systems
• Abstract \(\beta\)-expansions and ultimately periodic representations
• Atomic surfaces, tilings and coincidence. I: Irreducible case
• On the complexity of algebraic numbers. II: Continued fractions
• From number systems to shift radix systems
• Ergodic properties of signed binary expansions
• Nombres normaux applications aux fonctions pseudoaleatoires
• Les suites à spectre vide et la répartition modulo 1
• Markov subshifts and realization of \(\beta\)-expansions
• The alternating greedy expansion and applications to computing digit expansions from left-to-right in cryptography
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• On minimal expansions in redundant number systems: Algorithms and quantitative analysis
• Self-similar lattice tilings
• Pisot substitutions and Rauzy fractals
• On the numeration system of Ostrowski
• The scenery flow for geometric structures on the torus: The linear setting
• Disk-like self-affine tiles in \(\mathbb{R}^2\)
• On canonical number systems
• Bounded remainder sets
• Generalized radix representations and dynamical systems. I
• Geometric representation of substitutions of Pisot type
• An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation
• Analysis of linear combination algorithms in cryptography
• EPISTURMIAN WORDS: SHIFTS, MORPHISMS AND NUMERATION SYSTEMS
• Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable
• Digital sum moments and substitutions
• Finite beta-expansions
• Purely periodic 𝛽-expansions with Pisot unit base
• JACOBI MATRICES WITH RANDOM POTENTIALS TAKING FINITELY MANY VALUES
• Representations for real numbers and their ergodic properties
• Suites uniformément denses, moyennes trigonométriques, fonctions pseudo-aléatoires
• On theβ-expansions of real numbers
• Geometric theory of unimodular Pisot substitutions
• Best simultaneous diophantine approximations of Pisot numbers and Rauzy fractals
• SUR LE BÊTA-DÉVELOPPEMENT DE 1 DANS LE CORPS DES SÉRIES FORMELLES
• Nombres algébriques et substitutions
• Systems of Numeration
• Développement en base $\theta$, répartition modulo un de la suite $(x\theta \sp n)$, n$\ge 0$, langages codés et $\theta$-shift
• Substitutional dynamical systems, Bratteli diagrams and dimension groups
• Mesures spectrales associées à certaines suites arithmétiques
• On the one-sided boundedness of discrepancy function of the sequence {nα}
• Irregularities of uniform distribution
• Canonical number systems in imaginary quadratic fields
• On Periodic Expansions of Pisot Numbers and Salem Numbers
• Self-Similar Sets 5. Integer Matrices and Fractal Tilings of ℝ n
• Non singular transformations and spectral analysis of measures
• Generalized Rudin-Shapiro sequences
• Représentation géométrique de suites de complexité $2n+1$
• Suites à spectre vide et suites pseudo-aléatoires
• The spectrum of dynamical systems arising from substitutions of constant length
• Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity
• Sur la mesure spectrale de certaines suites arithmétiques
• Arithmetic construction of sofic partitions of hyperbolic toral automorphisms
• Geometric realizations of substitutions
• Numeration systems and Markov partitions from self similar tilings
• Beta-integers as natural counting systems for quasicrystals
• The Sum-of-Digits Function for Complex Bases
• Conjugates of Beta-Numbers and the Zero-Free Domain for a Class of Analytic Functions
• The zeta function of the beta transformation
• Systems of finite rank
• Dynamics of self-similar tilings
• Adding machines and wild attractors

This page was built for publication: Dynamical directions in numeration