scholarly journals The Pisot conjecture for -substitutions

2016 ◽  
Vol 38 (2) ◽  
pp. 444-472 ◽  
Author(s):  
MARCY BARGE
Keyword(s):  
Dynamical System ◽  
Discrete Spectrum ◽  
Minimal Polynomial ◽  
Pisot Number ◽  
Companion Matrix ◽  
Pisot Numbers ◽  
Almost Everywhere ◽  
One To One

We prove the Pisot conjecture for$\unicode[STIX]{x1D6FD}$-substitutions: if$\unicode[STIX]{x1D6FD}$is a Pisot number, then the tiling dynamical system$(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$associated with the$\unicode[STIX]{x1D6FD}$-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.

Pisot Numbers from {0, 1}-Polynomials

2010 ◽  
Vol 53 (1) ◽  
pp. 140-152
Author(s):  
Keshav Mukunda
Keyword(s):  
Unit Disk ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Pisot Number ◽  
Open Unit ◽  
Closed Unit Disk ◽  
Pisot Numbers ◽  
Salem Numbers ◽  
Salem Number ◽  
Newman Polynomials

AbstractA Pisot number is a real algebraic integer greater than 1, all of whose conjugates lie strictly inside the open unit disk; a Salem number is a real algebraic integer greater than 1, all of whose conjugate roots are inside the closed unit disk, with at least one of them of modulus exactly 1. Pisot numbers have been studied extensively, and an algorithm to generate them is well known. Our main result characterises all Pisot numbers whose minimal polynomial is derived from a Newman polynomial — one with {0, 1}-coefficients — and shows that they form a strictly increasing sequence with limit (1 + √5)/2. It has long been known that every Pisot number is a limit point, from both sides, of sequences of Salem numbers. We show that this remains true, from at least one side, for the restricted sets of Pisot and Salem numbers that are generated by Newman polynomials.

scholarly journals On purely periodic beta-expansions of Pisot numbers

2002 ◽  
Vol 166 ◽  
pp. 183-207 ◽  
Author(s):  
Yuki Sano
Keyword(s):  
Natural Extension ◽  
Irreducible Polynomial ◽  
Main Tool ◽  
Pisot Number ◽  
Fractal Boundary ◽  
Pisot Numbers ◽  
Dimensional Domain

AbstractWe characterize numbers having purely periodic β-expansions where β is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a d-dimensional domain with a fractal boundary.

An extension of the Cobham-Semënov Theorem

2000 ◽  
Vol 65 (1) ◽  
pp. 201-211 ◽  
Author(s):  
Alexis Bès
Keyword(s):  
Characteristic Polynomial ◽  
Minimal Polynomial ◽  
Pisot Numbers ◽  
Numeration Systems

AbstractLet θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

scholarly journals ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

On ℤ-Modules of Algebraic Integers

2009 ◽  
Vol 61 (2) ◽  
pp. 264-281 ◽  
Author(s):  
J. P. Bell ◽  
K. G. Hare
Keyword(s):  
Algebraic Integer ◽  
Full Rank ◽  
Pisot Number ◽  
List Type ◽  
Algebraic Integers ◽  
Pisot Numbers ◽  
Cyclotomic Numbers ◽  
Special Cases ◽  
Fixed Order ◽  
Fixed Positive Integer

Abstract. Let q be an algebraic integer of degree d ≥ 2. Consider the rank of the multiplicative subgroup of ℂ* generated by the conjugates of q. We say q is of full rank if either the rank is d − 1 and q has norm ±1, or the rank is d. In this paper we study some properties of ℤ[q] where q is an algebraic integer of full rank. The special cases of when q is a Pisot number and when q is a Pisot-cyclotomic number are also studied. There are four main results.(1)If q is an algebraic integer of full rank and n is a fixed positive integer, then there are only finitely many m such that disc `ℤ[qm]´ = disc `ℤ[qn]´.(2)If q and r are algebraic integers of degree d of full rank and ℤ[qn] = ℤ[rn] for infinitely many n, then either q = ωr′ or q = Norm(r)2/dω/r′ , where r ′ is some conjugate of r and ω is some root of unity.(3)Let r be an algebraic integer of degree at most 3. Then there are at most 40 Pisot numbers q such that ℤ[q] = ℤ[r].(4)There are only finitely many Pisot-cyclotomic numbers of any fixed order.

scholarly journals A combinatorial approach to products of Pisot substitutions

2015 ◽  
Vol 36 (6) ◽  
pp. 1757-1794 ◽  
Author(s):  
VALÉRIE BERTHÉ ◽  
JÉRÉMIE BOURDON ◽  
TIMO JOLIVET ◽  
ANNE SIEGEL
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Continued Fraction ◽  
Combinatorial Approach ◽  
Pisot Numbers ◽  
Algorithmic Framework ◽  
Multidimensional Continued Fraction ◽  
Finite Products ◽  
Dynamical Properties ◽  
Coincidence Conditions

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

On Devaney chaotic generalized shift dynamical systems

2013 ◽  
Vol 50 (4) ◽  
pp. 509-522 ◽  
Author(s):  
Fatemah Shirazi ◽  
Javad Sarkooh ◽  
Bahman Taherkhani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Point ◽  
Periodic Points ◽  
List Type ◽  
Topologically Transitive ◽  
Generalized Shift ◽  
One To One ◽  
Shift Dynamical System ◽  
Infinite Sequences

In the following text we prove that in a generalized shift dynamical system (XГ, σφ) for infinite countable Г and discrete X with at least two elements the following statements are equivalent: the dynamical system (XГ, σφ) is chaotic in the sense of Devaneythe dynamical system (XГ, σφ) is topologically transitivethe map φ: Г → Г is one to one without any periodic point.Also for infinite countable Г and finite discrete X with at least two elements (XГ, σφ) is exact Devaney chaotic, if and only if φ: Г → Г is one to one and φ: Г → Г has niether periodic points nor φ-backwarding infinite sequences.

scholarly journals On the lines passing through two conjugates of a Salem number

2008 ◽  
Vol 144 (1) ◽  
pp. 29-37 ◽  
Author(s):  
ARTŪRAS DUBICKAS ◽  
CHRIS SMYTH
Keyword(s):  
Algebraic Number ◽  
General Position ◽  
Imaginary Axis ◽  
Pisot Number ◽  
Pisot Numbers ◽  
Parallel Lines ◽  
Salem Number ◽  
Pass Through

AbstractWe show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d2/2]-d+2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ $\overline{\Q}$ of degree d are in general position if and only if α has 2 real conjugates, d-2 complex conjugates, no three distinct conjugates of α lie on a line and any two lines that pass through two distinct conjugates of α are non-parallel, except for d/2-1 lines parallel to the imaginary axis. Our main result asserts that the conjugates of any Salem number are in general position. We also ask two natural questions about conjugates of Pisot numbers which lead to the equation α1+α2=α3+α4 in distinct conjugates of a Pisot number. The Pisot number $\al_1\,{=}\,(1+\sqrt{3+2\sqrt{5}})/2$ shows that this equation has such a solution.

scholarly journals On Flows Spectrum on Closed Trio of Contours with Uniform Load

2018 ◽  
Vol 11 (1) ◽  
pp. 260 ◽  
Author(s):  
Alexander Pavlovich Buslaev ◽  
Alexander Gennadjevich Tatashev ◽  
Marina Victorovna Yashina
Keyword(s):  
Dynamical System ◽  
Conflict Resolution ◽  
Discrete Spectrum ◽  
Arbitrary Number ◽  
Priority Rule ◽  
Uniform Load ◽  
Remote Points

Considered dynamical system is a flow of clusters with the same length $l$ on contours of unit lengthconnected in polar-remote points into closed chain.When clusters move trough common node, the left-priority rule of conflict resolution works.In the paper it is shown that in the case of chain consisted third contours the dynamical system has a spectrum of velocity and mode periodicity consisted on not more two components.Distribution of spectrum in dependence on load $l$ is developed.Hypothesis on discrete spectrum in the case of arbitrary number of contours are formulated.

scholarly journals Bounded complexity, mean equicontinuity and discrete spectrum

2019 ◽  
Vol 41 (2) ◽  
pp. 494-533 ◽  
Author(s):  
WEN HUANG ◽  
JIAN LI ◽  
JEAN-PAUL THOUVENOT ◽  
LEIYE XU ◽  
XIANGDONG YE
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Invariant Measure ◽  
Discrete Spectrum ◽  
Ergodic Theory ◽  
Topological Dynamics ◽  
Topological Dynamical System ◽  
The Mean ◽  
Bounded Complexity

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

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