Dichotomy and Measures on Limit Sets of Anosov Groups

Source abstract

Abstract Let $G$ 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.