Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

This paper studies tilings related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic beta-expansion. Special focus is given to some quadratic examples

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(3) (2005), #A01 A NOTE ON LINEAR RECURRENT MAHLER NUMBERS

Arealnumberβ>1 is said to satisfy Property (F) if every non-negative number of Z[β −1] has a finite β-expansion. Such numbers are Pisot numbers. Using results of Laurent we prove under technical hypothesis that if G is the linear numeration system canonically associated with a number β satisfying (F), if H is a linear recurrence sequence and (nj)j is an unbounded sequence of natural integers such that lim sup nj = ∞, then the real number 0.(Hn0)G(Hn1)G ·· · does not belong to Q(β). This gives a new family of irrational numbers of Mahler’s type.

Propriétés topologiques et combinatoires des échelles de numération

Topological and combinatorial properties of dynamical systems called odometers and arising from number systems are investigated. First, a topological classification is obtained. Then a rooted tree describing the carries in the addition of 1 is introduced and extensively studied. It yields a description of points of discontinuity and a notion of low scale, which is helpful in producing examples of what the dynamics of an odometer can look like. Density of the orbits is also discussed

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper