### Abstract

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

Original language | English (US) |
---|---|

Pages (from-to) | 323-338 |

Number of pages | 16 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 33 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*33*(3), 323-338. https://doi.org/10.1016/S0246-0203(97)80094-5

}

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 33, no. 3, pp. 323-338. https://doi.org/10.1016/S0246-0203(97)80094-5

**Slow entropy type invariants and smooth realization of commuting measure-preserving transformations.** / Katok, Anatoly; Thouvenot, Jean Paul.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

AU - Katok, Anatoly

AU - Thouvenot, Jean Paul

PY - 1997/1/1

Y1 - 1997/1/1

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.

UR - http://www.scopus.com/inward/record.url?scp=0031498135&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031498135&partnerID=8YFLogxK

U2 - 10.1016/S0246-0203(97)80094-5

DO - 10.1016/S0246-0203(97)80094-5

M3 - Article

AN - SCOPUS:0031498135

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

SN - 0246-0203

IS - 3

ER -