Symmetry of entropy in higher rank diagonalizable actions and measure classification

Manfred Einsiedler; Elon Lindenstrauss

doi:10.3934/jmd.2018016

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2018, Volume 13: 163-185. Doi: 10.3934/jmd.2018016 open access

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Symmetry of entropy in higher rank diagonalizable actions and measure classification

Manfred Einsiedler^1, and
Elon Lindenstrauss^2,

1.
D-Math, ETH Zürich, Rämistrasse 101, CH-8092 Zürich, Switzerland
2.
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 9190401, Israel

In memory of Roy Adler

Received: August 10, 2017

Revised: December 11, 2017

Published: December 2018

ME: Supported by the SNF (Grant 200021-152819).
EL: Supported by ISF grant 891/15.

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Abstract

An important consequence of the theory of entropy of $ \mathbb{Z}$-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.

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Mathematics Subject Classification: Primary: 37D40; Secondary: 37A35, 37A45.

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References

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