Combinatorial and probabilistic properties of systems of numeration

Abstract

Let $G=(G_{n})_{n}$ be a strictly increasing sequence of positive integers with $G_{0}=1$. We study the system of numeration defined by this sequence by looking at the corresponding compactification ${\mathcal{K}}_{G}$ of $\mathbb{N}$ and the extension of the addition-by-one map ${\it\tau}$ on ${\mathcal{K}}_{G}$ (the ‘odometer’). We give sufficient conditions for the existence and uniqueness of ${\it\tau}$-invariant measures on ${\mathcal{K}}_{G}$ in terms of combinatorial properties of $G$.