  We elaborate notions of integration over the space of arcs factorized by the
natural $C^*$-action and over the space of non-parametrized arcs (branches).
There are offered two motivic versions of the zeta function of the classical
monodromy transformation of a germ of an analytic function on a smooth space.
We indicate a direct formula which connects the naive motivic zeta function of
J. Denef and F. Loeser with the classical monodromy zeta function.

Algebraic Geometry