Abstract
Let G be a connected semisimple real algebraic group and Γ a Zariski dense Anosov subgroup of G with respect to a minimal parabolic subgroup P. Let N be the maximal horospherical subgroup of G given by the unipotent radical of P. We describe the N-ergodic decompositions of all Burger–Roblin measures as well as the A-ergodic decompositions of all Bowen–Margulis–Sullivan measures on ΓG. As a consequence, we obtain the following refinement of the main result of [17]: the space of all non-trivial N-invariant ergodic and P°-quasi-invariant Radon measures on ΓG, up to constant multiples, is homeomorphic to ℝrank G−1 × {1, …, k} where k is the number of P°-minimal subsets in ΓG.
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Acknowledgements
We would like to thank Michael Hochman for helpful conversations, especially for telling us about the reference [11]. We also thank the referee for reading the manuscript carefully and making a useful suggestion.
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Lee and Oh respectively supported by the NSF grants DMS-1926686 (via the Institute for Advanced Study) and DMS-1900101.
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Lee, M., Oh, H. Ergodic decompositions of geometric measures on Anosov homogeneous spaces. Isr. J. Math. (2023). https://doi.org/10.1007/s11856-023-2560-2
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DOI: https://doi.org/10.1007/s11856-023-2560-2