Slow entropy type invariants and smooth realization of commuting measure-preserving transformations. (English) Zbl 0884.60009

Katok, Anatole; Thouvenot, Jean-Paul

Ann. Inst. Henri Poincaré, Probab. Stat. 33, No. 3, 323-338 (1997).

Summary: 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 \(\mathbb{Z}^2\) actions with zero one-dimensional entropies in all (including irrational) directions which do not allow either of the above realizations.

MSC:

60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
28D05	Measure-preserving transformations

Keywords:

measure-preserving actions; exponential growth rate; invariant measure