Algebraic entropy of amenable group actions

Let R be a ring, let G be an amenable group and let RG be a crossed product. The goal of this paper is to construct, starting with a suitable additive function L on the category of left modules over R, an additive function on a subcategory of the category of left modules over RG, which coincides with the whole category if L(RR)<. This construction can be performed using a dynamical invariant associated with the original function L, called algebraic L-entropy. We apply our results to two classical problems on group rings: the stable finiteness and the zero-divisors conjectures.