Horospherical invariant measures and a rank dichotomy for Anosov groups

Or Landesberg; Minju Lee; Elon Lindenstrauss; Hee Oh

doi:10.3934/jmd.2023009

Article Contents

2023, Volume 19: 331-362. Doi: 10.3934/jmd.2023009

This volume Previous Article The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups Next Article Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom

Horospherical invariant measures and a rank dichotomy for Anosov groups

1.
Department of Mathematics, Yale University, New Haven, CT 06520, USA
2.
Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

MINJU LEE
MINJU LEE

Received: September 15, 2021

Revised: October 31, 2022

Published online: May 4, 2023

OL: Partially supported by ISF-Moked grant 2095/19. EL: Partially supported by ERC 2020 grant no. 833423. HO: Partially supported by the NSF grant 0003086.

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Abstract

Let $ G = \prod_{i = 1}^{\mathsf r} G_i $ be a product of simple real algebraic groups of rank one and $ \Gamma $ an Anosov subgroup of $ G $ with respect to a minimal parabolic subgroup. For each $ \mathsf v $ in the interior of a positive Weyl chamber, let $ \mathscr R_ \mathsf v\subset \Gamma\backslash G $ denote the Borel subset of all points with recurrent $ \exp (\mathbb R_+ \mathsf v) $-orbits. For a maximal horospherical subgroup $ N $ of $ G $, we show that the $ N $-action on $ {\mathscr R}_ \mathsf v $ is uniquely ergodic if $ \mathsf r = \operatorname{rank}(G)\le 3 $ and $ \mathsf v $ belongs to the interior of the limit cone of $ \Gamma $, and that there exists no $ N $-invariant Radon measure on $ \mathscr R_ \mathsf v $ otherwise.

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Mathematics Subject Classification: Primary: 37A17, 37A40, 22E40; Secondary: 22F30.

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[1]	M. Babillot, On the classification of invariant measures for horosphere foliations on nilpotent covers of negatively curved manifolds, in, Random Walks and Geometry, 319–335, Walter de Gruyter, Berlin, 2004.
[2]	M. Babillot and F. Ledrappier, Geodesic paths and horocycle flow on abelian covers, in Lie Groups and Ergodic Theory (Mumbai, 1996), 1–32, Tata Inst. Fund. Res. Stud. Math., vol. 14, Bombay, 1998.
[3]	Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47. doi: 10.1007/PL00001613.
[4]	B. H. Bowditch, Geometrical finiteness for hyperbolic groups, J. Funct. Anal., 113 (1993), 245-317. doi: 10.1006/jfan.1993.1052.
[5]	E. Breuillard, Geometry of locally compact groups of polynomial growth and shape of large balls, Groups Geom. Dyn., 8 (2014), 669-732. doi: 10.4171/GGD/244.
[6]	M. Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J., 61 (1990), 779-803. doi: 10.1215/S0012-7094-90-06129-0.
[7]	M. Burger, Intersection, the Manhattan curve, and Patterson-Sullivan theory in rank 2, Internat. Math. Res. Not., 1993,217–225. doi: 10.1155/S1073792893000236.
[8]	M. Burger, O. Landesberg, M. Lee and H. Oh, The Hopf–Tsuji–Sullivan dichotomy in higher rank and applications to Anosov subgroups, J. Mod. Dyn., 19 (2023), 301-330. doi: 10.3934/jmd.2023008.
[9]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, vol. 580, Springer-Verlag, Berlin-New York, 1977.
[10]	S. G. Dani, Invariant measures and minimal sets of horospherical flows, Invent. Math., 64 (1981), 357-385. doi: 10.1007/BF01389173.
[11]	A. H. Dooley and K. Jarrett, Non-singular $\mathbb{Z}^d$ -actions: An ergodic theorem over rectangles with application to the critical dimensions, Ergodic Theory Dynam. Syst., 41 (2021), 3722-3739. doi: 10.1017/etds.2020.116.
[12]	S. Edwards, M. Lee and H. Oh, Anosov groups: Local mixing, counting, and equidistribution, preprint, arXiv: 2003.14277, 2020, to appear in Geometry & Topology.
[13]	M. Fraczyk and T. Gelander, Infinite volume and injectivity radius, Ann. of Math., 197 (2023), 389-421. doi: 10.4007/annals.2023.197.1.6.
[14]	H. Furstenberg, The unique ergodicity of the horocycle flow, in Recent Advances in Topological Dynamics (Proc. Conf. Yale Univ., 1972; in honor of G. A. Hedlund), pp. 95–115, Lecture Notes in Math., vol. 318, Springer, Berlin, 1973.
[15]	O. Guichard and A. Wienhard, Anosov representations: Domains of discontinuity and applications, Invent. Math., 190 (2012), 357-438. doi: 10.1007/s00222-012-0382-7.
[16]	Y. Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101 (1973), 333-397.
[17]	Y. Guivarc'h and A. Raugi, Actions of large semigroups and random walks on isometric extensions of boundaries, Ann. Sci. École Norm. Sup. (4), 40 (2007), 209-249. doi: 10.1016/j.ansens.2007.01.003.
[18]	M. Hochman, A ratio ergodic theorem for multiparameter non-singular actions, J. Eur. Math. Soc., 12 (2010), 365-383. doi: 10.4171/JEMS/201.
[19]	K. Jarrett, An ergodic theorem for nonsingular actions of the Heisenberg groups, Trans. Amer. Math. Soc., 372 (2019), 5507-5529. doi: 10.1090/tran/7750.
[20]	F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114. doi: 10.1007/s00222-005-0487-3.
[21]	O. Landesberg, Horospherically invariant measures and finitely generated Kleinian groups, J. Mod. Dyn., 17 (2021), 337-352. doi: 10.3934/jmd.2021012.
[22]	O. Landesberg and E. Lindenstrauss, On Radon measures invariant under horospherical flows on geometrically infinite quotients, Int. Math. Res. Not., 2022, 11602–11641. doi: 10.1093/imrn/rnab024.
[23]	E. Le Donne and S. Rigot, Besicovitch covering property on graded groups and applications to measure differentiation, J. Reine Angew. Math., 750 (2019), 241-297. doi: 10.1515/crelle-2016-0051.
[24]	F. Ledrappier, Invariant measures for the stable foliation on negatively curved periodic manifolds, Ann. Inst. Fourier, 58 (2008), 85-105. doi: 10.5802/aif.2345.
[25]	F. Ledrappier and O. Sarig, Invariant measures for the horocycle flow on periodic hyperbolic surfaces, Israel J. Math., 160 (2007), 281-315. doi: 10.1007/s11856-007-0064-0.
[26]	M. Lee and H. Oh, Invariant measures for horospherical actions and Anosov groups, preprint, arXiv: 2008.05296, 2020, to appear in Int. Math. Res. Not. doi: 10.1093/imrn/rnac262.
[27]	M. Lee and H. Oh, Ergodic decompositions of geometric measures on Anosov homogeneous spaces, preprint, arXiv: 2010.11337, 2020, to appear in Israel J. Math.
[28]	A. Mohammadi and H. Oh, Classification of joinings for Kleinian groups, Duke Math. J., 165 (2016), 2155-2223. doi: 10.1215/00127094-3476807.
[29]	G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Princeton Univ. Press, 1973.
[30]	H. Oh and W. Pan, Local mixing and invariant measures for horospherical subgroups on abelian covers, Int. Math. Res. Not., 2019, 6036–6088. doi: 10.1093/imrn/rnx292.
[31]	J.-F. Quint, Mesures de Patterson-Sullivan en rang supérieur, Geom. Funct. Anal., 12 (2002), 776-809. doi: 10.1007/s00039-002-8266-4.
[32]	J.-F. Quint, L'indicateur de croissance des groupes de Schottky, Ergodic Theory Dynam. Syst., 23 (2003), 249-272. doi: 10.1017/S0143385702001268.
[33]	M. Ratner, On Raghunathan's measure conjecture, Ann. Math., 134 (1991), 545-607. doi: 10.2307/2944357.
[34]	T. Roblin, Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr., 95 (2003), vi+96 pp. doi: 10.24033/msmf.408.
[35]	A. Sambarino, Quantitative properties of convex representations, Comment. Math. Helv., 89 (2014), 443-488. doi: 10.4171/CMH/324.
[36]	O. Sarig, Invariant Radon measures for horocycle flows on abelian covers, Invent. Math., 157 (2004), 519-551. doi: 10.1007/s00222-004-0357-4.
[37]	O. Sarig, The horocycle flow and the Laplacian on hyperbolic surfaces of infinite genus, Geom. Funct. Anal., 19 (2010), 1757-1812. doi: 10.1007/s00039-010-0048-9.
[38]	B. Schapira, A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds, Confluentes Math., 8 (2016), 165-174. doi: 10.5802/cml.29.
[39]	K. Schmidt, Unique ergodicity and related problems, in Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978), pp. 188–198, Lecture Notes in Math., vol. 729, Springer, Berlin, 1979.
[40]	D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 171-202.
[41]	J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. doi: 10.1016/0021-8693(72)90058-0.
[42]	W. A. Veech, Unique ergodicity of horospherical flows, Amer. J. Math., 99 (1977), 827-859. doi: 10.2307/2373868.
[43]	D. Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math., 210 (2015), 467-507. doi: 10.1007/s11856-015-1258-5.