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Arithmetic Dynamics
, 2002
"... This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepres ..."
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Cited by 5 (1 self)
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This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepreserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: • Betaexpansions, i.e., the radix expansions in noninteger bases; • “Rotational ” expansions which arise in the problem of encoding of irrational rotations of the circle; • Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodictheoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create “redundant” representations (those whose space of “digits ” is a priori larger than necessary)
PERIODIC UNIQUE BETAEXPANSIONS: THE SHARKOVSKIĬ ORDERING
"... ABSTRACT. Let β ∈ (1, 2). Each x ∈ [0, β−1] can be represented in the form x = εkβ −k, k=1 where εk ∈ {0, 1} for all k (a βexpansion of x). If β> 1+√5 2, then, as is well known, there 1 always exist x ∈ (0, β−1) which have a unique βexpansion. In the present paper we study (purely) periodic unique ..."
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Cited by 4 (4 self)
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ABSTRACT. Let β ∈ (1, 2). Each x ∈ [0, β−1] can be represented in the form x = εkβ −k, k=1 where εk ∈ {0, 1} for all k (a βexpansion of x). If β> 1+√5 2, then, as is well known, there 1 always exist x ∈ (0, β−1) which have a unique βexpansion. In the present paper we study (purely) periodic unique βexpansions and show that for each n ≥ 2 there exists βn ∈ [ 1+ √ 5 2, 2) such that there are no unique periodic βexpansions of smallest period n for β ≤ βn and at least one such expansion for β> βn. Furthermore, we prove that βk < βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps. 1. HISTORY OF THE PROBLEM AND FORMULATION OF RESULTS This paper continues the line of research related to the combinatorics of representations of real numbers in noninteger bases ([12, 13, 15, 21]). 1 Let β ∈ (1, 2) be our parameter and let x ∈ Iβ: = [0,]. Then x has at least one represenβ−1 tation of the form (1.1) x = πβ(ε1, ε2,...): = εkβ −k, εk ∈ {0, 1}, k=1 (use, e.g., the greedy algorithm) which we call a βexpansion of x and write x ∼ (ε1, ε2,...)β. Let us recall some key results regarding βexpansions. Firstly, if 1 < β < G: = 1+√5, then 2 each x ∈ � � 1 0, has a continuum of βexpansions [12]. On the other hand, for any β> G, β−1 there exist infinitely many x which have a unique βexpansion (see [9, 13]), although almost all x ∈ Iβ still have a continuum of βexpansions [21]. More specifically, put x ∼ (010101...)β = 1 β2. Then both x and βx have a unique β
ON UNIVOQUE PISOT NUMBERS
, 2007
"... We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = ∑ n≥1 snβ−n, with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the s ..."
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Cited by 3 (1 self)
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We study Pisot numbers β ∈ (1, 2) which are univoque, i.e., such that there exists only one representation of 1 as 1 = ∑ n≥1 snβ−n, with sn ∈ {0, 1}. We prove in particular that there exists a smallest univoque Pisot number, which has degree 14. Furthermore we give the smallest limit point of the set of univoque Pisot numbers.
Univoque numbers and an avatar of ThueMorse
"... Univoque numbers are real numbers λ> 1 such that the number 1 admits a unique expansion in base λ, i.e., a unique expansion 1 = ∑ j≥0 ajλ−(j+1) , with aj ∈ {0,1,..., ⌈λ ⌉ − 1} for every j ≥ 0. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called ..."
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Univoque numbers are real numbers λ> 1 such that the number 1 admits a unique expansion in base λ, i.e., a unique expansion 1 = ∑ j≥0 ajλ−(j+1) , with aj ∈ {0,1,..., ⌈λ ⌉ − 1} for every j ≥ 0. A variation of this definition was studied in 2002 by Komornik and Loreti, together with sequences called admissible sequences. We show how a 1983 study of the first author gives both a result of Komornik and Loreti on the smallest admissible sequence on the set {0,1,...,b}, and a result of de Vries and Komornik (2007) on the smallest univoque number belonging to the interval (b,b + 1), where b is any positive integer. We also prove that this last number is transcendental. An avatar of the ThueMorse sequence, namely the fixed point beginning in 3 of the morphism 3 → 31, 2 → 30, 1 → 03, 0 → 02, occurs in a “universal ” manner.
EXPANSIONS IN NONINTEGER BASES
"... Representations1 of real numbers in noninteger bases were introduced by Rényi [18] and first studied by Rényi and by Parry [17]. Let first β be an integer greater than 1. Then any number x ∈ [0,1) can be represented ..."
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Representations1 of real numbers in noninteger bases were introduced by Rényi [18] and first studied by Rényi and by Parry [17]. Let first β be an integer greater than 1. Then any number x ∈ [0,1) can be represented
UNIQUE EXPANSIONS OF REAL NUMBERS MARTIJN DE VRIES
, 2008
"... Abstract. It was discovered some years ago that there exist noninteger real numbers q> 1 for which only one sequence (ci) of integers ci ∈ [0, q) satisfies the equality P∞ i=1 ciq−i = 1. The set of such “univoque numbers ” has a rich topological structure, and its study revealed a number of unexpec ..."
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Abstract. It was discovered some years ago that there exist noninteger real numbers q> 1 for which only one sequence (ci) of integers ci ∈ [0, q) satisfies the equality P∞ i=1 ciq−i = 1. The set of such “univoque numbers ” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper we consider for each fixed q> 1 the set Uq of real numbers x having a unique representation of the form P∞ i=1 ciq−i = x with integers ci belonging to [0, q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set U ′ q consisting of all sequences (ci) of integers ci ∈ [0, q) such that P∞ i=1 ciq−i ∈ Uq. We determine the numbers r> 1 for which the map q ↦ → U ′ q (defined on (1, ∞)) is constant in a neighborhood of r and the numbers q> 1 for which U ′ q is a subshift or a subshift of finite type.
Sturmian words, βshifts, and transcendence ∗
, 2008
"... Consider the minimal βshift containing the shift space generated by given Sturmian word. In this paper we characterize such β and investigate its combinatorial, dynamical and topological properties and prove that such β are transcendental numbers. 1. Introduction. Sturmian ..."
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Consider the minimal βshift containing the shift space generated by given Sturmian word. In this paper we characterize such β and investigate its combinatorial, dynamical and topological properties and prove that such β are transcendental numbers. 1. Introduction. Sturmian
A PROPERTY OF ALGEBRAIC UNIVOQUE NUMBERS
, 2007
"... Abstract. Consider the set U of real numbers q ≥ 1 for which only one sequence (ci) of integers 0 ≤ ci ≤ q satisfies the equality P∞ i=1 ciq−i = 1. In this note we show that the set of algebraic numbers in U is dense in the closure U of U. 1. ..."
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Abstract. Consider the set U of real numbers q ≥ 1 for which only one sequence (ci) of integers 0 ≤ ci ≤ q satisfies the equality P∞ i=1 ciq−i = 1. In this note we show that the set of algebraic numbers in U is dense in the closure U of U. 1.
Dynamics for βshifts and Diophantine
, 2007
"... ∗Provisional—final page numbers to be inserted when paper edition is published ..."
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∗Provisional—final page numbers to be inserted when paper edition is published