We provide a new proof of the fact that the horospherical group acting on the frame bundle of a hyperbolic manifold admits a unique invariant ergodic measure (up to multiplicative constants) supported on the set of frames whose orbit under the geodesic flow comes back infinitely often in a compact set. This result is known, but our proof is more direct and much shorter.
@article{CML_2016__8_1_165_0,
author = {Barbara Schapira},
title = {A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds},
journal = {Confluentes Mathematici},
pages = {165--174},
publisher = {Institut Camille Jordan},
volume = {8},
number = {1},
year = {2016},
doi = {10.5802/cml.29},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.5802/cml.29/}
}
TY - JOUR AU - Barbara Schapira TI - A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds JO - Confluentes Mathematici PY - 2016 SP - 165 EP - 174 VL - 8 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.29/ DO - 10.5802/cml.29 LA - en ID - CML_2016__8_1_165_0 ER -
%0 Journal Article %A Barbara Schapira %T A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds %J Confluentes Mathematici %D 2016 %P 165-174 %V 8 %N 1 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.29/ %R 10.5802/cml.29 %G en %F CML_2016__8_1_165_0
Barbara Schapira. A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds. Confluentes Mathematici, Tome 8 (2016) no. 1, pp. 165-174. doi : 10.5802/cml.29. https://cml.centre-mersenne.org/articles/10.5802/cml.29/
[1] Geodesic paths and horocycle flow on abelian covers, Lie groups and ergodic theory (Mumbai, 1996) (Tata Inst. Fund. Res. Stud. Math.), Volume 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 1-32
[2] Horocycle flow on geometrically finite surfaces, Duke Math. J., Volume 61 (1990) no. 3, pp. 779-803
[3] A short proof of the unique ergodicity of horocyclic flows, Ergodic theory (Contemp. Math.), Volume 485, Amer. Math. Soc., Providence, RI, 2009, pp. 85-89
[4] Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., Volume 47 (1978) no. 2, pp. 101-138
[5] Geometrically finite groups, Patterson-Sullivan measures and Ratner’s rigidity theorem, Invent. Math., Volume 99 (1990) no. 3, pp. 601-626
[6] The unique ergodicity of the horocycle flow, Recent advances in topological dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Springer, Berlin, 1973, p. 95-115. Lecture Notes in Math., Vol. 318
[7] A ratio ergodic theorem for multiparameter non-singular actions, J. Eur. Math. Soc. (JEMS), Volume 12 (2010) no. 2, pp. 365-383
[8] Distribution of orbits in of a finitely generated group of , Amer. J. Math., Volume 136 (2014) no. 6, pp. 1497-1542
[9] Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) (2003) no. 95, vi+96 pages
[10] Invariant Radon measures for horocycle flows on abelian covers, Invent. Math., Volume 157 (2004) no. 3, pp. 519-551
[11] On quasi-invariant transverse measures for the horospherical foliation of a negatively curved manifold, Ergodic Theory Dynam. Systems, Volume 24 (2004) no. 1, pp. 227-255
[12] Mixing of frame flow for rank one locally symmetric spaces and measure classification, Isr. J. Math. (2015)
Cité par Sources :