Sur Le Bêta-Développement De 1 Dans Le Corps Des Séries Formelles

Abstract

Let β be a fixed element of 𝔽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk [Formula: see text]. Any formal power series in [Formula: see text] is expanded in this way into dβ(f) = (ai(X))i≥1, where [Formula: see text]. The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).

Rational digit systems over finite fields and Christol's Theorem

“…The P/Q-digit expansions we have defined here are not the same as the β-expansions of formal Laurent series over F q that were introduced independently in [13,22]. To be more precise, if β = P/Q, then the β-expansion of α ∈ D(0, 1) is given by…”

Dynamical directions in numeration

“…The β-expansions have also been extended to the case of finite fields independently by Scheicher [289], as well as Hbaib and Mkaouar [163]. Let F((x −1 )) be the field of formal Laurent series over F and denote by |·| some absolute value.…”

“…In [163], the "representation of 1", which is defined in terms of an analogue of the β-transformation, is studied.…”

ON THE PERIOD OF β-EXPANSION OF PISOT OR SALEM SERIES OVER 𝔽q((x-1))

“…So, if z, w ∈ F q ((x −1 )), then we have d β (z +w) = d β (z)+d β (w) digitwise and if c ∈ F * q , then d β (cz) = cd β (z). In the case of the field of formal series, the following theorems were proved independently by Hbaib-Mkaouar and Scheicher in the statements [12] and [7]. In the papers [9] and [10], metric results are established and the relation to continued fractions is studied.…”