Expanding measures: Random walks and rigidity on homogeneous spaces
Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423000567/type/journal_article |
| _version_ | 1797787964708749312 |
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| author | Roland Prohaska Cagri Sert Ronggang Shi |
| author_facet | Roland Prohaska Cagri Sert Ronggang Shi |
| author_sort | Roland Prohaska |
| collection | DOAJ |
| description | Let G be a real Lie group,
$\Lambda <G$
a lattice and
$H\leqslant G$
a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures
$\mu $
on H and, applying recent work of Eskin–Lindenstrauss, prove that
$\mu $
-stationary probability measures on
$G/\Lambda $
are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on
$G/\Lambda $
satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in
$G/\Lambda $
for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting. |
| first_indexed | 2024-03-13T01:29:27Z |
| format | Article |
| id | doaj.art-cb87515223544623894d8c071888cb29 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-13T01:29:27Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-cb87515223544623894d8c071888cb292023-07-04T09:18:31ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.56Expanding measures: Random walks and rigidity on homogeneous spacesRoland Prohaska0Cagri Sert1Ronggang Shi2Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland; E-mail:Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland; E-mail:Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, No.2005 Songhu Road, Shanghai, 200438, China; E-mail:Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu $ -stationary probability measures on $G/\Lambda $ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on $G/\Lambda $ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda $ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.https://www.cambridge.org/core/product/identifier/S2050509423000567/type/journal_article60B1522F3060G5037A4428A80 |
| spellingShingle | Roland Prohaska Cagri Sert Ronggang Shi Expanding measures: Random walks and rigidity on homogeneous spaces Forum of Mathematics, Sigma 60B15 22F30 60G50 37A44 28A80 |
| title | Expanding measures: Random walks and rigidity on homogeneous spaces |
| title_full | Expanding measures: Random walks and rigidity on homogeneous spaces |
| title_fullStr | Expanding measures: Random walks and rigidity on homogeneous spaces |
| title_full_unstemmed | Expanding measures: Random walks and rigidity on homogeneous spaces |
| title_short | Expanding measures: Random walks and rigidity on homogeneous spaces |
| title_sort | expanding measures random walks and rigidity on homogeneous spaces |
| topic | 60B15 22F30 60G50 37A44 28A80 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423000567/type/journal_article |
| work_keys_str_mv | AT rolandprohaska expandingmeasuresrandomwalksandrigidityonhomogeneousspaces AT cagrisert expandingmeasuresrandomwalksandrigidityonhomogeneousspaces AT ronggangshi expandingmeasuresrandomwalksandrigidityonhomogeneousspaces |