A short proof of unique ergodicity of horospherical foliations on infinite volume hyperbolic manifolds
Confluentes Mathematici, Volume 8 (2016) no. 1, pp. 165-174.

We provide a new proof of the fact that the horospherical group N<G=SO o (d,1) acting on the frame bundle ΓG 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/cml.29
Classification: 22E40, 22D40, 28D15, 37A17, 37A25
Keywords: unique ergodicity, horospherical group, frame bundle, in nite volume hyperbolic manifolds
Barbara Schapira 1

1 I.R.M.A.R. UMR CNRS 6625, UFR de mathématiques, Campus de Beaulieu, 263 avenue du Général Leclerc, CS 74205 35042 RENNES Cédex, France
@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, Volume 8 (2016) no. 1, pp. 165-174. doi : 10.5802/cml.29. https://cml.centre-mersenne.org/articles/10.5802/cml.29/

[1] Martine Babillot; François Ledrappier 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] Marc Burger Horocycle flow on geometrically finite surfaces, Duke Math. J., Volume 61 (1990) no. 3, pp. 779-803

[3] Yves Coudene 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] S. G. Dani Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math., Volume 47 (1978) no. 2, pp. 101-138

[5] L. Flaminio; R. J. Spatzier Geometrically finite groups, Patterson-Sullivan measures and Ratner’s rigidity theorem, Invent. Math., Volume 99 (1990) no. 3, pp. 601-626

[6] Harry Furstenberg 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] Michael Hochman A ratio ergodic theorem for multiparameter non-singular actions, J. Eur. Math. Soc. (JEMS), Volume 12 (2010) no. 2, pp. 365-383

[8] François Maucourant; Barbara Schapira Distribution of orbits in 2 of a finitely generated group of SL (2,), Amer. J. Math., Volume 136 (2014) no. 6, pp. 1497-1542

[9] Thomas Roblin Ergodicité et équidistribution en courbure négative, Mém. Soc. Math. Fr. (N.S.) (2003) no. 95, vi+96 pages

[10] Omri Sarig Invariant Radon measures for horocycle flows on abelian covers, Invent. Math., Volume 157 (2004) no. 3, pp. 519-551

[11] Barbara Schapira 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] Dale Winter Mixing of frame flow for rank one locally symmetric spaces and measure classification, Isr. J. Math. (2015)

Cited by Sources: