Expansions in multiple bases over general alphabets

Zou, Yuru; Komornik, Vilmos; Lü, Jian

doi:10.48550/arxiv.2102.10051

Cited by 3 publications

(3 citation statements)

References 20 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 96%

Spectrum, algebraicity and normalization in alternate bases

Charlier¹,

Cisternino²,

Masáková³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Gathering both, the notion of Cantor real bases was introduced by Charlier and the author in a recent work [4]. Note that these type of representations involving more than one base simultaneously and independently aroused the interest of mathematicians [2,4,8,9,12].…”

Section: Introductionmentioning

confidence: 99%

Combinatorial properties of lazy expansions in Cantor real bases

Cisternino¹

2022

Preprint

View full text Add to dashboard Cite

The lazy algorithm for a real base β is generalized to the setting of Cantor bases β = (βn) n∈N introduced recently by Charlier and the author. To do so, let x β be the greatest real number that has a β-representation a0a1a2 • • • such that each letter an belongs to {0, . . . , ⌈βn⌉ − 1}. This paper is concerned with the combinatorial properties of the lazy β-expansions, which are defined when x β < +∞. As an illustration, Cantor bases following the Thue-Morse sequence are studied and a formula giving their corresponding value of x β is proved. First, it is shown that the lazy β-expansions are obtained by "flipping" the digits of the greedy β-expansions. Next, a Parry-like criterion characterizing the sequences of non-negative integers that are the lazy β-expansions of some real number in (x β − 1, x β ] is proved. Moreover, the lazy β-shift is studied and in the particular case of alternate bases, that is the periodic Cantor bases, an analogue of Bertrand-Mathis' theorem in the lazy framework is proved: the lazy β-shift is sofic if and only if all quasi-lazy β (i) -expansions of x β (i) − 1 are ultimately periodic, where β (i) is the i-th shift of the alternate base β.

show abstract

“…Representations involving more than one base have recently gained momentum, as shown by the five simultaneous and independent works [4,5,18,20,27]. In particular, these papers all present a generalization of Parry's theorem to their respective frameworks.…”

Section: Introductionmentioning

confidence: 99%

Dynamical behavior of alternate base expansions

Charlier¹,

Cisternino²,

Dajani

2021

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

We generalize the greedy and lazy $\beta $ -transformations for a real base $\beta $ to the setting of alternate bases ${\boldsymbol {\beta }}=(\beta _0,\ldots ,\beta _{p-1})$ , which were recently introduced by the first and second authors as a particular case of Cantor bases. As in the real base case, these new transformations, denoted $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ respectively, can be iterated in order to generate the digits of the greedy and lazy ${\boldsymbol {\beta }}$ -expansions of real numbers. The aim of this paper is to describe the measure-theoretical dynamical behaviors of $T_{{\boldsymbol {\beta }}}$ and $L_{{\boldsymbol {\beta }}}$ . We first prove the existence of a unique absolutely continuous (with respect to an extended Lebesgue measure, called the p-Lebesgue measure) $T_{{\boldsymbol {\beta }}}$ -invariant measure. We then show that this unique measure is in fact equivalent to the p-Lebesgue measure and that the corresponding dynamical system is ergodic and has entropy $({1}/{p})\log (\beta _{p-1}\cdots \beta _0)$ . We give an explicit expression of the density function of this invariant measure and compute the frequencies of letters in the greedy ${\boldsymbol {\beta }}$ -expansions. The dynamical properties of $L_{{\boldsymbol {\beta }}}$ are obtained by showing that the lazy dynamical system is isomorphic to the greedy one. We also provide an isomorphism with a suitable extension of the $\beta $ -shift. Finally, we show that the ${\boldsymbol {\beta }}$ -expansions can be seen as $(\beta _{p-1}\cdots \beta _0)$ -representations over general digit sets and we compare both frameworks.

show abstract

Expansions in multiple bases over general alphabets

Cited by 3 publications

References 20 publications

Spectrum, algebraicity and normalization in alternate bases

Spectrum, algebraicity and normalization in alternate bases

Combinatorial properties of lazy expansions in Cantor real bases

Dynamical behavior of alternate base expansions

Contact Info

Product

Resources

About