Dichotomy and measures on limit sets of Anosov groups

TL;DR

This paper establishes a criterion for the support of conformal measures on limit sets of Anosov groups, proving uniqueness and extending classical dichotomy results to higher rank groups with applications to measure conjectures.

Contribution

It introduces a higher rank analogue of the Hopf-Tsuji-Sullivan dichotomy and characterizes conformal measures support in terms of critical dimension for Anosov subgroups.

Findings

Support of conformal measures characterized by critical dimension

Uniqueness of conformal measures at critical dimension

Extension of Ahlfors measure conjecture to Anosov subgroups

Abstract

Let $G$ be a connected semisimple real algebraic group. For any Zariski dense Anosov subgroup $\Gamma <G$, we show that a $\Gamma$-conformal measure is supported on the limit set of $\Gamma$ if and only if its "dimension" is $\Gamma$-critical. This implies the uniqueness of a $\Gamma$-conformal measure for each critical dimension. We deduce this from a higher rank analogue of the Hopf-Tsuji-Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups.