Motivic infinite cyclic covers

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (under certain finiteness conditions) an element in the Grothendieck ring K0(VarCμˆ), which we call motivic infinite cyclic cover, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fiber of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.