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Soient $U$ et $V$ deux systèmes de numération de Bertrand, $\alpha$ et $\beta$ deux $\beta$-nombres multiplicativement indépendants tels que $L\left(U\right)=L\left(\alpha \right)$ et $L\left(V\right)=L\left(\beta \right)$, et $E$ un sous-ensemble de $\mathbb{N}$. Si $E$ est $U$-reconnaissable et $V$-reconnaissable alors $E$ est une réunion finie de progressions arithmétiques.

In this paper we prove the decidability of the HD0L ultimate periodicity problem.

The seminal theorem of Cobham has given rise during the last 40 years to a lot of work about non-standard numeration systems and has been extended to many contexts. In this paper, as a result of fifteen years of improvements, we obtain a complete and general version for the so-called substitutive sequences. Let $\alpha$ and $\beta$ be two multiplicatively independent Perron numbers. Then a sequence $x\in A^{\mathbb{N}}$, where $A$ is a finite alphabet, is both $\alpha$-substitutive and $\beta$-substitutive if and only if $x$ is ultimately periodic....

We study the convergence of the ergodic averages $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ T^{u_k}$ where ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ is a bounded sequence and ${\left(u_k\right)}_{k\in \mathbb{N}}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ for some $\delta \<1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.