  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 $\phi$, and study some of their basic properties. Some
elementary connections between these notions and motivic integration theory are
also provided.
