Invariant Radon measures for Unipotent flows and products of Kleinian groups

Mohammadi, Amir; Oh, Hee

doi:10.1090/proc/13840

Invariant Radon measures for Unipotent flows and products of Kleinian groups
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by Amir Mohammadi and Hee Oh PDF

Proc. Amer. Math. Soc. 146 (2018), 1469-1479 Request permission

Abstract:

Let $G= \textrm {PSL}_2(\mathbb {F})$ where $\mathbb {F}= \mathbb {R} , \mathbb {C}$, and consider the space $Z=(\Gamma _1 \times \Gamma _2)\backslash (G\times G)$ where $\Gamma _1<G$ is a co-compact lattice and $\Gamma _2<G$ is a geometrically finite discrete Zariski dense subgroup. For a horospherical subgroup $N$ of $G$, we classify all ergodic, conservative, invariant Radon measures on $Z$ for the diagonal $N$-action.

References

Jon Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol. 50, American Mathematical Society, Providence, RI, 1997. MR 1450400, DOI 10.1090/surv/050
Martine Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, Random walks and geometry, Walter de Gruyter, Berlin, 2004, pp. 319–335. MR 2087786
M. Bachir Bekka and Matthias Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, vol. 269, Cambridge University Press, Cambridge, 2000. MR 1781937, DOI 10.1017/CBO9780511758898
Yves Benoist and Hee Oh, Fuchsian groups and compact hyperbolic surfaces, Enseign. Math. 62 (2016), no. 1-2, 189–198. MR 3605815, DOI 10.4171/LEM/62-1/2-11
Yves Benoist and Jean-François Quint, Mesures stationnaires et fermés invariants des espaces homogènes, Ann. of Math. (2) 174 (2011), no. 2, 1111–1162 (French, with English and French summaries). MR 2831114, DOI 10.4007/annals.2011.174.2.8
Yves Benoist and Jean-François Quint, Stationary measures and invariant subsets of homogeneous spaces (III), Ann. of Math. (2) 178 (2013), no. 3, 1017–1059. MR 3092475, DOI 10.4007/annals.2013.178.3.5
Marc Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J. 61 (1990), no. 3, 779–803. MR 1084459, DOI 10.1215/S0012-7094-90-06129-0
Harry Furstenberg, The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 95–115. MR 0393339
Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR 797411, DOI 10.1515/9783110844641
L. Flaminio and R. J. Spatzier, Geometrically finite groups, Patterson-Sullivan measures and Ratner’s rigidity theorem, Invent. Math. 99 (1990), no. 3, 601–626. MR 1032882, DOI 10.1007/BF01234433
François Ledrappier, Invariant measures for the stable foliation on negatively curved periodic manifolds, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 1, 85–105 (English, with English and French summaries). MR 2401217
François Ledrappier and Omri Sarig, Invariant measures for the horocycle flow on periodic hyperbolic surfaces, Israel J. Math. 160 (2007), 281–315. MR 2342499, DOI 10.1007/s11856-007-0064-0
G. A. Margulis and G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994), no. 1-3, 347–392. MR 1253197, DOI 10.1007/BF01231565
Amir Mohammadi and Hee Oh, Classification of joinings for Kleinian groups, Duke Math. J. 165 (2016), no. 11, 2155–2223. MR 3536991, DOI 10.1215/00127094-3476807
H. Oh and W. Pan, Local mixing and invariant measures for horospherical subgroups on abelian covers, Preprint, arXiv:1701.02772
Marina Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), no. 3, 545–607. MR 1135878, DOI 10.2307/2944357
Thomas Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) 95 (2003), vi+96 (French, with English and French summaries). MR 2057305, DOI 10.24033/msmf.408
Daniel J. Rudolph, Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold, Ergodic Theory Dynam. Systems 2 (1982), no. 3-4, 491–512 (1983). MR 721736, DOI 10.1017/S0143385700001735
Omri Sarig, Invariant Radon measures for horocycle flows on abelian covers, Invent. Math. 157 (2004), no. 3, 519–551. MR 2092768, DOI 10.1007/s00222-004-0357-4
Omri M. Sarig, Unique ergodicity for infinite measures, Proceedings of the International Congress of Mathematicians. Volume III, Hindustan Book Agency, New Delhi, 2010, pp. 1777–1803. MR 2827866
Barbara Schapira, Lemme de l’ombre et non divergence des horosphères d’une variété géométriquement finie, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 939–987 (French, with English and French summaries). MR 2111017
Nimish A. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Lie groups and ergodic theory (Mumbai, 1996) Tata Inst. Fund. Res. Stud. Math., vol. 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 229–271. MR 1699367
Dale Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math. 210 (2015), no. 1, 467–507. MR 3430281, DOI 10.1007/s11856-015-1258-5
Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR 776417, DOI 10.1007/978-1-4684-9488-4