We classify the measures on $SL(k,\mathbb{R}) / SL(k,\mathbb{Z})$ which are invariant and ergodic under the action of the group $A$ of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood’s conjecture has Hausdorff dimension zero.
Manfred Einsiedler 1 ; Anatole Katok 2 ; Elon Lindenstrauss 1
@article{10_4007_annals_2006_164_513,
author = {Manfred Einsiedler and Anatole Katok and Elon Lindenstrauss},
title = {Invariant measures and the set of exceptions to {Littlewood{\textquoteright}s} conjecture},
journal = {Annals of mathematics},
pages = {513--560},
year = {2006},
volume = {164},
number = {2},
doi = {10.4007/annals.2006.164.513},
mrnumber = {2247967},
zbl = {1109.22004},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.164.513/}
}
TY - JOUR AU - Manfred Einsiedler AU - Anatole Katok AU - Elon Lindenstrauss TI - Invariant measures and the set of exceptions to Littlewood’s conjecture JO - Annals of mathematics PY - 2006 SP - 513 EP - 560 VL - 164 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.164.513/ DO - 10.4007/annals.2006.164.513 LA - en ID - 10_4007_annals_2006_164_513 ER -
%0 Journal Article %A Manfred Einsiedler %A Anatole Katok %A Elon Lindenstrauss %T Invariant measures and the set of exceptions to Littlewood’s conjecture %J Annals of mathematics %D 2006 %P 513-560 %V 164 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.164.513/ %R 10.4007/annals.2006.164.513 %G en %F 10_4007_annals_2006_164_513
Manfred Einsiedler; Anatole Katok; Elon Lindenstrauss. Invariant measures and the set of exceptions to Littlewood’s conjecture. Annals of mathematics, Tome 164 (2006) no. 2, pp. 513-560. doi: 10.4007/annals.2006.164.513
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