Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

We define invariants for measure-preserving actions of discrete amenable groups which characterize various subexponential rates of growth for the number of "essential" orbits similarly to the way entropy of the action characterizes the exponential growth rate. We obtain above estimates for these invariants for actions by diffeomorphisms of a compact manifold (with a Borel invariant measure) and, more generally, by Lipschitz homeomorphisms of a compact metric space of finite box dimension. We show that natural cutting and stacking constructions alternating independent and periodic concatenation of names produce ℤ² actions with zero one-dimensional entropies in all (including irrational) directions which do not allow either of the above realizations.