Geodesic
Integration over Spaces of Nonparametrized Arcs and Motivic Versions of the Monodromy Zeta Function
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 71-82.

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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. 
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     title = {Integration over {Spaces} of {Nonparametrized} {Arcs} and {Motivic} {Versions} of the {Monodromy} {Zeta} {Function}},
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S. M. Gusein-Zade; I. Luengo; A. Melle-Hernández. Integration over Spaces of Nonparametrized Arcs and Motivic Versions of the Monodromy Zeta Function. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 71-82. http://geodesic.mathdoc.fr/item/TM_2006_252_a7/

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