Substitution Dynamical Systems - Spectral Analysis

Queffélec, Martine

doi:10.1007/978-3-642-11212-6

Cited by 346 publications

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“…this sequence R is called Rudin-Shapiro sequence. Its elements have the following properties: (Their proofs and further formulas can be found in [11]. )…”

Section: Upper Bound For Small Values Of W α Uniformly In α For the Rmentioning

confidence: 99%

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On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

Dartyge

Gyarmati

Sárközy

2018

Uniform Distribution Theory

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“…this sequence R is called Rudin-Shapiro sequence. Its elements have the following properties: (Their proofs and further formulas can be found in [11]. )…”

Section: Upper Bound For Small Values Of W α Uniformly In α For the Rmentioning

confidence: 99%

“…In this section our goal is to give a lower bound for W α for the Legendre symbol sequence E p p¡1 defined as in (11):…”

Section: Lower Bound For W α For the Legendre Symbol Constructionmentioning

confidence: 99%

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

Dartyge

Gyarmati

Sárközy

2018

Uniform Distribution Theory

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“…Importantly, such dynamical systems connected to all fields described above: they appear as a return mapping of the expanding flow onto the contracting manifold for hyperbolic toral automorphisms with a unique expanding direction [20] (in relation with dynamical systems and Markov partitions); they allow one to generate infinite sequences with interesting combinatorial features (in relation with combinatorics of words); they provide a construction of points sets for quasicristals (in relation with theoretical physics); they also allow one to describe non unique greedy expansions of reals in non integer basis. From their self-similar structure, their symbolic dynamical systems are naturally generated by substitutions [49,78]. Therefore, substitutions appear to be ubiquitous to describe self-repetitive mathematical processes.…”

Section: Substitutions Among Mathematics and Computer Sciencementioning

confidence: 99%

Machines, Computations, and Universality

2015

Lecture Notes in Computer Science

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In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties of the substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems.

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“…One can check that the substitution above is primitive, proper and aperiodic and therefore defines a minimal Cantor system (Z , R) (see [28] for an introduction to substitution systems). The space Z is the subset of {a, b, c} Z which consists of those sequences which can be uniquely written as concatenations of words σ k (a), σ k (b) and σ k (c) for any k > 0, e.g.…”

Section: Example Of Non-conjugate Flow-orbit Equivalent Systemsmentioning

confidence: 99%

Flow–orbit equivalence for minimal Cantor systems

Kosek

Ormes

Rudolph

2008

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385707000703 How to cite this article: WOJCIECH KOSEK, NICHOLAS ORMES and DANIEL J. RUDOLPH (2008). Flow-orbit equivalence for minimal Cantor systems.Abstract. This paper is about flow-orbit equivalence, a topological analog of even Kakutani equivalence. In addition to establishing many basic facts about this relation, we characterize the conjugacies of induced systems that can be extended to a floworbit equivalence. We also describe the relationship between flow-orbit equivalence and a distortion function of an orbit equivalence. We show that, if the distortion of an orbit equivalence is zero, then it is in fact a flow-orbit equivalence, and that the converse is true up to a conjugation by an element of the full group.

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Substitution Dynamical Systems - Spectral Analysis

Cited by 346 publications

References 0 publications

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

On Irregularities of Distribution of Binary Sequences Relative to Arithmetic Progressions, II (Constructive Bounds)

Machines, Computations, and Universality

Flow–orbit equivalence for minimal Cantor systems

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