arXiv 2603.286932026-03-30math.GTmath.DSmath.GR

向量值horofunction边界与Patterson-Sullivan测度

Vector-valued horofunction boundaries and Patterson--Sullivan measures

Simons Laufer Mathematical Sciences Institute（西蒙斯劳弗数学科学研究所）；Department of Mathematics, University of Wisconsin-Madison（威斯康星大学麦迪逊分校数学系）

Dongryul M. Kim, Andrew Zimmer

备注：: 28 pages, Comments welcome!

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AI总结：

本文研究了高维空间中基于向量值horofunction边界的Patterson-Sullivan测度的存在性和唯一性，引入了影子概念并证明了其唯一性和遍历性结果。

摘要：

在高维空间中，Patterson-Sullivan测度支持在部分旗 manifold 上有良好的理论研究，但其存在性和唯一性证明是一个难题。本文发展了支持在关联对称空间的向量值horofunction边界上的Patterson-Sullivan测度理论，其中存在性是直接的。我们还引入了该紧致化中的影子概念，并建立了影子引理。对于横截群，我们证明了唯一性和遍历性结果。

英文摘要：

In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.

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