2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71297Let $(X,x)$ be a germ of real or complex analytic space and $\mathcal{A}_{(X,x)}$ the space of germs of arcs on $(X,x)$. Let us consider $F_{x}: (X,x) \to (Y,y)$ a germ of a morphism and denote by $\mathcal{F}_{x}: \mathcal{A}_{(X,x)} \to \mathcal{A}_{(Y,y)}$ the induced morphism at the level of arcs. In this paper, we try to emphasize the analogies between the metric or local topological properties of $F_{x}$ and those of $\mathcal{F}_{x}$. We then define the notions of Nash sequence of multiplicities, Nash sequence of Hilbert-Samuel functions and Nash sequence of diagram of initial exponents of $X$ along an arc $ϕ$, and study some of their basic properties. Some elementary connections between these notions and motivic integration theory are also provided.45 pages, frenchAlgebraic GeometryCategory Theory14B05; 32S05; 32S99text