Dichotomy and measures on limit sets of Anosov groups
Dichotomy and measures on limit sets of Anosov groups
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发表时间：
2022-03
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通讯作者：
Min Ho Lee;H. Oh
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作者：
Min Ho Lee;H. Oh
设G是连通的半单真实的代数群.对任意Zebriki稠密Anosov子群$\Gamma<G$，证明了$\Gamma$-共形测度在$\Gamma$的极限集上成立当且仅当它的“维数“是$\Gamma$-临界的.这意味着一个$\Gamma$-共形措施的每个关键尺寸的唯一性。我们从最大对角作用的Hopf-Tsuji-Sullivan二分法的更高等级类似物中推导出这一点。其他应用包括类似的Ahlfors措施猜想Anosov子群。
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.