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Non-concentration property of Patterson–Sullivan measures for Anosov subgroups
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 45 / Issue 3 / March 2025
- Published online by Cambridge University Press:
- 18 September 2024, pp. 884-893
- Print publication:
- March 2025
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- Article
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Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup
$\Gamma <G$ with respect to a parabolic subgroup 
$P_\theta $, we prove that any 
$\Gamma $-Patterson–Sullivan measure charges no mass on any proper subvariety of 
$G/P_\theta $. More generally, we prove that for a Zariski dense 
$\theta $-transverse subgroup 
$\Gamma <G$, any 
$(\Gamma , \psi )$-Patterson–Sullivan measure charges no mass on any proper subvariety of 
$G/P_\theta $, provided the 
$\psi $-Poincaré series of 
$\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.
