Integration over Spaces of Nonparametrized Arcs and Motivic Versions of the Monodromy Zeta Function

Notions of integration of motivic type over the space of arcs factorized by the natural $\mathbb C^*$-action and over the space of nonparametrized arcs (branches) are developed. As an application, two motivic versions of the zeta function of the classical monodromy transformation of a germ of an analytic function on $\mathbb C^d$ are given that correspond to these notions. Another key ingredient in the construction of these motivic versions of the zeta function is the use of the so-called power structure over the Grothendieck ring of varieties introduced by the authors.