Non-uniform Expansions of Real Numbers

Neunhäuserer, Jörg

doi:10.1007/s00009-021-01723-7

Cited by 9 publications

(5 citation statements)

References 38 publications

(42 reference statements)

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“…In section 2.2. we see that multiple base expansions of x ∈ [0, 1], whose properties have been investigated recently (e.g. [20] and [23]), are generated by such a random dynamical system. In particular, we give a sufficient condition to guarantee 'almost all' expansions actually converge to x ∈ [0, 1].…”

Section: Preliminariesmentioning

confidence: 97%

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

Suzuki

2024

Nonlinearity

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We consider an independent and identically distributed (i.i.d.) random dynamical system of simple linear transformations on the unit interval T β ( x ) = β x (mod 1), x ∈ [ 0 , 1 ] , β > 0, which are the so-called beta-transformations. For such a random dynamical system, including the case that it is generated by uncountably many maps, we give an explicit formula for the density function of a unique stationary measure under the assumption that the random dynamics is expanding in mean. As an application, in the case that the random dynamics is generated by finitely many maps and the maps are chosen according to a Bernoulli measure, we show that the density function is analytic as a function of parameter in the Bernoulli measure and give its derivative explicitly. Furthermore, for a non-i.i.d. random dynamical system of beta-transformations, we also give an explicit formula for the random densities of a unique absolutely continuous invariant measure under a certain strong expanding condition or under the assumption that the maps randomly chosen are close to the beta-transformation for a non-simple number in the sense of parameter β.

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Section: Preliminariesmentioning

confidence: 97%

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

Suzuki

2024

Nonlinearity

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“…The same framework was introduced simultaneously in [6]. Moreover, other kinds of expansions with multiple real bases were also recently studied, see [23,25,28].…”

Section: Cantor Real Bases and Alternate Basesmentioning

confidence: 99%

Alternate Base Numeration Systems

Charlier

2023

Lecture Notes in Computer Science

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Alternate base numeration systems generalize real base numeration systems as defined by Renyi. Real numbers are represented using a finite number of bases periodically. Such systems naturally appear when considering linear numeration systems without a dominant root. As it happens, many classical results generalize to these numeration systems with multiple bases but some don't. This is a survey of the work done so far concerning combinatorial, algebraic and dynamical aspects. This study has been led in collaboration with several co-authors :

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“…Representations of real numbers involving more than one base simultaneously and independently aroused the interest of other mathematicians [4,14,16,23]. But so far, most of the research was concentrated on the combinatorial properties of these representations as in [6] and not on their dynamical or algebraic properties which are studied respectively in [7] and in this paper.…”

Section: Introductionmentioning

confidence: 96%

Spectrum, algebraicity and normalization in alternate bases

Charlier¹,

Cisternino²,

Masáková³

et al. 2022

Preprint

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The first aim of this article is to give information about the algebraic properties of alternate bases β = (β0, . . . , βp−1) determining sofic systems. We show that a necessary condition is that the product δ = p−1 i=0 βi is an algebraic integer and all of the bases β0, . . . , βp−1 belong to the algebraic field Q(δ). On the other hand, we also give a sufficient condition: if δ is a Pisot number and β0, . . . , βp−1 ∈ Q(δ), then the system associated with the alternate base β = (β0, . . . , βp−1) is sofic. The second aim of this paper is to provide an analogy of Frougny's result concerning normalization of real bases representations. We show that given an alternate base β = (β0, . . . , βp−1) such that δ is a Pisot number and β0, . . . , βp−1 ∈ Q(δ), the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. The spectrum of a real number δ > 1 and an alphabet A ⊂ Z was introduced by Erdős et al. For our purposes, we use a generalized concept with δ ∈ C and A ⊂ C and study its topological properties.

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Non-uniform Expansions of Real Numbers

Abstract: We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.

Cited by 9 publications

References 38 publications

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

Absolutely continuous invariant measures for random dynamical systems of beta-transformations

Alternate Base Numeration Systems

Spectrum, algebraicity and normalization in alternate bases

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