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On the entropy theory of finitely-generated nilpotent group actions

Published online by Cambridge University Press:  06 November 2002

VALENTIN YA. GOLODETS  and
SERGEY D. SINEL'SHCHIKOV
VALENTIN YA. GOLODETS
Affiliation:
Institute for Low Temperature Physics and Engineering, Ukrainian National Academy of Sciences, 47 Lenin Ave, 61103 Kharkov, Ukraine (e-mail: [email protected]; [email protected])
SERGEY D. SINEL'SHCHIKOV
Affiliation:
Institute for Low Temperature Physics and Engineering, Ukrainian National Academy of Sciences, 47 Lenin Ave, 61103 Kharkov, Ukraine (e-mail: [email protected]; [email protected])
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Abstract

The entropy for actions of a finitely-generated nilpotent group G is investigated. The Pinsker algebra of such actions is described explicitly. The systems with completely positive entropy are shown to have a sort of ‘asymptotic independence property’, just as in the case of \mathbb{Z}^d-actions. The invariant partitions are used to prove that the property of completely positive entropy is equivalent to the property of the K-system (the property of the existence of special ‘good’ partitions). A complete spectral characterization of K-systems is given. A construction of examples of completely positive non-Bernoullian actions of general countable nilpotent groups and a class of solvable groups is presented.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 6 , December 2002 , pp. 1747 - 1771
Copyright
2002 Cambridge University Press

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