TY - JOUR
T1 - Slow entropy type invariants and smooth realization of commuting measure-preserving transformations
AU - Katok, Anatole
AU - Thouvenot, Jean Paul
N1 - Funding Information:
* Partially supported by NSF grant DMS-94 04061. ** The second author would like to express his gratitude to the Pennsylvania State University mathematics department for financial support and warm hospitality during his visit in January-February 1995 when the present paper was written.
PY - 1997
Y1 - 1997
N2 - 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.
AB - 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.
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U2 - 10.1016/S0246-0203(97)80094-5
DO - 10.1016/S0246-0203(97)80094-5
M3 - Article
AN - SCOPUS:0031498135
SN - 0246-0203
VL - 33
SP - 323
EP - 338
JO - Annales de l'institut Henri Poincare (B) Probability and Statistics
JF - Annales de l'institut Henri Poincare (B) Probability and Statistics
IS - 3
ER -