Power structure over the Grothendieck ring of maps

A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2+\ldots)^m$, where $a_i$, $i=1, 2,\ldots$, and $m$ are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of $\lambda$- and power structures over some Grothendieck rings of varieties. The main example is for the Grothendieck ring of maps of complex quasi-projective varieties. We describe two natural $\lambda$-structures on it which lead to the same power structure. We show that this power structure is effective. In the terms of this power structure we write some equations containing classes of Hilbert-Chow morphisms. We describe some generalizations of this construction for maps of varieties with some additional structures.