Dichotomy and Measures on Limit Sets of Anosov Groups

Abstract

AbstractLet $G$ be a connected semisimple real algebraic group. For a 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, answering the question posed in our earlier paper with Edwards [13]. We obtain this by proving 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.