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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.
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We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $$3$$ -manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents.more » « less
