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Harmonic and invariant measures on foliated spaces


Authors: Chris Connell and Matilde Martínez
Journal: Trans. Amer. Math. Soc. 369 (2017), 4931-4951
MSC (2010): Primary 37C40, 53C12, 58J65; Secondary 57R30, 37D40
DOI: https://doi.org/10.1090/tran/6811
Published electronically: March 6, 2017
MathSciNet review: 3632555
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Abstract: We consider the family of harmonic measures on a lamination $ \mathscr {L}$ of a compact space $ X$ by locally symmetric spaces $ L$ of noncompact type, i.e. $ L\cong \Gamma _L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $ G$-orbits, $ \hat {\mathscr {L}}$, which are right invariant under a minimal parabolic (Borel) subgroup $ B<G$. In the special case when $ G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $ \hat {\mathscr {L}}$ right invariant under two distinct minimal parabolics, and therefore all of $ G$, are in bijective correspondence with the holonomy invariant ones.


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Additional Information

Chris Connell
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

Matilde Martínez
Affiliation: Department of Mathematics, Universidad de la República, 2544 Montevideo, Uruguay

DOI: https://doi.org/10.1090/tran/6811
Received by editor(s): December 15, 2009
Received by editor(s) in revised form: October 29, 2012, and August 15, 2015
Published electronically: March 6, 2017
Additional Notes: The first author was supported by NSF grant DMS-0608643
The second author was supported by ANII, research grant FCE2007\underline{ }106.
Article copyright: © Copyright 2017 American Mathematical Society