Abstract

Adic systems are minimal symbolic dynamical systems that they are transversal to subshift of finite type. In a similar manner as the horocycle flows are transversal to the geodesic flows on surfaces of constant negative curvature. Let $(X,T)$ be an adic system and $Y$ be a closed non-empty subset of $X$. The induced map of $T$ on $Y$ is $WT(x)=Tn(x)(x)$ where $x\; \in \; Y$ and $n(x)$ is the return time of $x$ in $Y$. If the system $(Y,WT)$ is topologically equivalent to an adic system, we say that $(Y,WT)$ is an sub-adic system. We study a classification under bi-Lipschitz maps of the sub-adic systems. The tool used in this task is the computation of the spectrum of the dimensions of the Poincaré recurrences of the system. We also consider a similar problem in the geometrical setting. Under some hypothesis the adic systems are realized as (i.e. semi-conjugate to) translations on the $d$-dimensional torus, where $d\; \ge \; 1$.