Integration over spaces of non-parametrized arcs and motivic versions of the monodromy zeta function

Abstract

We elaborate notions of integration over the space of arcs factorized by the natural $C^{\ast }$-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.