Geodesic
Expanding measures: Random walks and rigidity on homogeneous spaces
Forum of Mathematics, Sigma, Tome 11 (2023)

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Let G be a real Lie group, $\Lambda $ 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. 
@article{10_1017_fms_2023_56,
     author = {Roland Prohaska and Cagri Sert and Ronggang Shi},
     title = {Expanding measures: {Random} walks and rigidity on homogeneous spaces},
     journal = {Forum of Mathematics, Sigma},
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     year = {2023},
     doi = {10.1017/fms.2023.56},
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     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.56/}
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Roland Prohaska; Cagri Sert; Ronggang Shi. Expanding measures: Random walks and rigidity on homogeneous spaces. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.56

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