SLOW ENTROPY TYPE INVARIANTS AND SMOOTH REALIZATION OF COMMUTING MEASURE-PRESERVING TRANSFORMATIONS

We define invariants for measure-preserving actions of discrete amenab le groups which characterize various subexponential Fates of growth fo r the number of ''essential'' orbits similarly to the way entropy of t he action characterizes the exponential growth rate. We obtain above e stimates for these invariants for actions by diffeomorphisms of a comp act manifold (with a Borel invariant measure) and, more generally, by Lipschitz homeomorphisms of a compact metric space of finite box dimen sion. We show that natural cutting and stacking constructions alternat ing independent and periodic concatenation of names produce Z(2) actio ns with zero one-dimensional entropies in all (including irrational) d irections which do not allow either of the above realizations.