TY - JOUR

T1 - The Fried Average Entropy and Slow Entropy for Actions of Higher Rank Abelian Groups

AU - Katok, Anatoly

AU - Katok, Svetlana

AU - Rodriguez Hertz, Federico Juan

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider two numerical entropy-type invariants for actions of ℤk, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. ℤ, k ≥ 2 actions on k + 1-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of ℤ actions the entropies of ergodic maximal rank actions take only countably many values. Our main result is the dichotomy that is best expressed under the assumption of weak mixing or, equivalently, no periodic factors: either both invariants vanish, or their values are bounded away from zero by universal constants. Furthermore, the lower bounds grow with dimension: for the first invariant (the Fried average entropy) exponentially, and for the second (the slow entropy) linearly.

AB - We consider two numerical entropy-type invariants for actions of ℤk, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. ℤ, k ≥ 2 actions on k + 1-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of ℤ actions the entropies of ergodic maximal rank actions take only countably many values. Our main result is the dichotomy that is best expressed under the assumption of weak mixing or, equivalently, no periodic factors: either both invariants vanish, or their values are bounded away from zero by universal constants. Furthermore, the lower bounds grow with dimension: for the first invariant (the Fried average entropy) exponentially, and for the second (the slow entropy) linearly.

UR - http://www.scopus.com/inward/record.url?scp=84906344895&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84906344895&partnerID=8YFLogxK

U2 - 10.1007/s00039-014-0284-5

DO - 10.1007/s00039-014-0284-5

M3 - Article

AN - SCOPUS:84906344895

VL - 24

SP - 1204

EP - 1228

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -