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Symmetry of entropy in higher rank diagonalizable actions and measure classification

In memory of Roy Adler

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

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


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