2018, 13: 163-185. doi: 10.3934/jmd.2018016

Symmetry of entropy in higher rank diagonalizable actions and measure classification

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

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

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.

Citation: Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016
References:
[1]

M. Einsiedler and A. Katok, Invariant measures on $ G/Γ$ for split simple Lie groups $ G$, Comm. Pure Appl. Math., 56 (2003), 1184-1221.  doi: 10.1002/cpa.10092.  Google Scholar

[2]

M. Einsiedler and A. Katok, Rigidity of measures - the high entropy case, and non-commuting foliations. Probability in mathematics, Israel J. Math., 148 (2005), 169-238.   Google Scholar

[3]

M. EinsiedlerA. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2), 164 (2006), 513-560.  doi: 10.4007/annals.2006.164.513.  Google Scholar

[4]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 155-241.  Google Scholar

[5]

M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups, Ann. of Math. (2), 181 (2015), 993-1031.  doi: 10.4007/annals.2015.181.3.3.  Google Scholar

[6]

M. Einsiedler and E. Lindenstrauss, Joinings of higher rank torus actions on homogeneous spaces, Publications Mathématiques de l'IHÉS, (2016), 39 pp. Google Scholar

[7]

M. Einsiedler, E. Lindenstrauss and A. Mohammadi, Diagonal actions in positive characteristic, (2017), 41pp, arXiv: 1705.10418, Google Scholar

[8]

M. EinsiedlerE. LindenstraussP. Michel and A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J., 148 (2009), 119-174.  doi: 10.1215/00127094-2009-023.  Google Scholar

[9]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in Ergodic Theory and Homogeneous Dynamics, 2017, in preparation, https://tbward0.wixsite.com/books/entropy. Google Scholar

[10]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011.  Google Scholar

[11]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.  Google Scholar

[12]

B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 593-637. doi: 10.1090/pspum/069/1858547.  Google Scholar

[13]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, 16 (1996), 751-778.  doi: 10.1017/S0143385700009081.  Google Scholar

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.   Google Scholar

[15]

E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), 163 (2006), 165-219.  doi: 10.4007/annals.2006.163.165.  Google Scholar

[16]

E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems, 21 (2001), 1481-1500.  doi: 10.1017/S0143385701001717.  Google Scholar

[17]

G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1991.  Google Scholar

[18]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.   Google Scholar

[19]

G. A. Margulis and G. M. Tomanov, Measure rigidity for almost linear groups and its applications, J. Anal. Math., 69 (1996), 25-54.   Google Scholar

[20]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.  Google Scholar

[21]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., 165 (1990), 229-309.  doi: 10.1007/BF02391906.  Google Scholar

[22]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[23]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $ p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382.  doi: 10.1215/S0012-7094-95-07710-2.  Google Scholar

show all references

References:
[1]

M. Einsiedler and A. Katok, Invariant measures on $ G/Γ$ for split simple Lie groups $ G$, Comm. Pure Appl. Math., 56 (2003), 1184-1221.  doi: 10.1002/cpa.10092.  Google Scholar

[2]

M. Einsiedler and A. Katok, Rigidity of measures - the high entropy case, and non-commuting foliations. Probability in mathematics, Israel J. Math., 148 (2005), 169-238.   Google Scholar

[3]

M. EinsiedlerA. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood's conjecture, Ann. of Math. (2), 164 (2006), 513-560.  doi: 10.4007/annals.2006.164.513.  Google Scholar

[4]

M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, in Homogeneous Flows, Moduli Spaces and Arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 155-241.  Google Scholar

[5]

M. Einsiedler and E. Lindenstrauss, On measures invariant under tori on quotients of semisimple groups, Ann. of Math. (2), 181 (2015), 993-1031.  doi: 10.4007/annals.2015.181.3.3.  Google Scholar

[6]

M. Einsiedler and E. Lindenstrauss, Joinings of higher rank torus actions on homogeneous spaces, Publications Mathématiques de l'IHÉS, (2016), 39 pp. Google Scholar

[7]

M. Einsiedler, E. Lindenstrauss and A. Mohammadi, Diagonal actions in positive characteristic, (2017), 41pp, arXiv: 1705.10418, Google Scholar

[8]

M. EinsiedlerE. LindenstraussP. Michel and A. Venkatesh, Distribution of periodic torus orbits on homogeneous spaces, Duke Math. J., 148 (2009), 119-174.  doi: 10.1215/00127094-2009-023.  Google Scholar

[9]

M. Einsiedler, E. Lindenstrauss and T. Ward, Entropy in Ergodic Theory and Homogeneous Dynamics, 2017, in preparation, https://tbward0.wixsite.com/books/entropy. Google Scholar

[10]

M. Einsiedler and T. Ward, Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, 259, Springer-Verlag London, Ltd., London, 2011.  Google Scholar

[11]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995.  Google Scholar

[12]

B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999), Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, 2001, 593-637. doi: 10.1090/pspum/069/1858547.  Google Scholar

[13]

A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems, 16 (1996), 751-778.  doi: 10.1017/S0143385700009081.  Google Scholar

[14]

E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. Math., 146 (2001), 259-295.   Google Scholar

[15]

E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2), 163 (2006), 165-219.  doi: 10.4007/annals.2006.163.165.  Google Scholar

[16]

E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems, 21 (2001), 1481-1500.  doi: 10.1017/S0143385701001717.  Google Scholar

[17]

G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1991.  Google Scholar

[18]

G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math., 116 (1994), 347-392.   Google Scholar

[19]

G. A. Margulis and G. M. Tomanov, Measure rigidity for almost linear groups and its applications, J. Anal. Math., 69 (1996), 25-54.   Google Scholar

[20]

M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York, 1972.  Google Scholar

[21]

M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math., 165 (1990), 229-309.  doi: 10.1007/BF02391906.  Google Scholar

[22]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. (2), 134 (1991), 545-607.  doi: 10.2307/2944357.  Google Scholar

[23]

M. Ratner, Raghunathan's conjectures for Cartesian products of real and $ p$-adic Lie groups, Duke Math. J., 77 (1995), 275-382.  doi: 10.1215/S0012-7094-95-07710-2.  Google Scholar

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