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

Conjugacy of unimodular Pisot substitutions subshifts to domain exchanges

We prove that any unimodular Pisot substitution subshift is measurably conjugate to a domain exchange in Euclidean spaces which factorizes onto a minimal rotation on a torus. This generalizes the pioneer works of Rauzy and Arnoux-Ito providing geometric realizations to any unimodular Pisot substitution without any additional combinatorial condition.Comment: 29 p. In this new version, a gap in the proof of the main theorem has been fixe

Combinatorics of Pisot Substitutions

Siirretty Doriast

Combinatorics of Multicompositions

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatorial properties of these multicompositions. In particular, we determine the count of multicompositions by number of all parts, number of positive parts, and number of zeros. Then, working from three types of compositions with restricted parts that are counted by the Fibonacci sequence, we find the sequences counting multicompositions with analogous restrictions. With these tools, we give combinatorial proofs of summation formulas for generalizations of the Jacobsthal and Pell sequences.Comment: 13 page