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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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Invariant Measures for Horospherical Actions and Anosov Groups
Abstract Let $$\Gamma $$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $$G$$. For a maximal horospherical subgroup $$N$$ of $$G$$, we show that the space of all non-trivial $NM$-invariant ergodic and $$A$$-quasi-invariant Radon measures on $$\Gamma \backslash G$$, up to proportionality, is homeomorphic to $${\mathbb {R}}^{\text {rank}\,G-1}$$, where $$A$$ is a maximal real split torus and $$M$$ is a maximal compact subgroup that normalizes $$N$$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures.
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- Award ID(s):
- 1900101
- PAR ID:
- 10418894
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnac262
