  Given an ergodic probability measure preserving dynamical system $\G\acts
(X,\mu)$, where $\G$ is a finitely generated countable group, we show that the
asymptotic growth of the number of finite models for the dynamics, in the sense
of sofic approximations, is an invariant of orbit equivalence. We then prove an
additivity formula for free products with amenable (possibly trivial)
amalgamation. In particular, we obtain purely combinatorial proofs of several
results in orbit equivalence theory.
