Sur quelques aspects de la g\'eom\'etrie de l'espace des arcs trac\'es sur un espace analytique

Let $(X,x)$ 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 $\varphi$, and study some of their basic properties. Some elementary connections between these notions and motivic integration theory are also provided.