Voir la notice de l'article provenant de la source Mathematical Sciences Publishers
In this paper we consider geodesic flow on finite-volume hyperbolic manifolds with non-empty totally geodesic boundary. We analyse the time for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the zeroth and first moments correspond to two cases of known identities for the orthospectrum. We also show that the second moment is given by the average time for the geodesic flow to hit the boundary. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case.
Bridgeman, Martin 1 ; Tan, Ser Peow 2
@article{GT_2014_18_1_a11,
author = {Bridgeman, Martin and Tan, Ser Peow},
title = {Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold},
journal = {Geometry & topology},
pages = {491--520},
publisher = {mathdoc},
volume = {18},
number = {1},
year = {2014},
doi = {10.2140/gt.2014.18.491},
url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2014.18.491/}
}
TY - JOUR AU - Bridgeman, Martin AU - Tan, Ser Peow TI - Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold JO - Geometry & topology PY - 2014 SP - 491 EP - 520 VL - 18 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/gt.2014.18.491/ DO - 10.2140/gt.2014.18.491 ID - GT_2014_18_1_a11 ER -
%0 Journal Article %A Bridgeman, Martin %A Tan, Ser Peow %T Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold %J Geometry & topology %D 2014 %P 491-520 %V 18 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/gt.2014.18.491/ %R 10.2140/gt.2014.18.491 %F GT_2014_18_1_a11
Bridgeman, Martin; Tan, Ser Peow. Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold. Geometry & topology, Tome 18 (2014) no. 1, pp. 491-520. doi : 10.2140/gt.2014.18.491. http://geodesic.mathdoc.fr/articles/10.2140/gt.2014.18.491/
[1] , The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993) 1139
[2] , The geometry of Teichmüller space via geodesic currents, Invent. Math. 92 (1988) 139
[3] , Orthospectra of geodesic laminations and dilogarithm identities on moduli space, Geom. Topol. 15 (2011) 707
[4] , , Distribution of intersection lengths of a random geodesic with a geodesic lamination, Ergodic Theory Dynam. Systems 27 (2007) 1055
[5] , , Hyperbolic volume of manifolds with geodesic boundary and orthospectra, Geom. Funct. Anal. 20 (2010) 1210
[6] , Chimneys, leopard spots and the identities of Basmajian and Bridgeman, Algebr. Geom. Topol. 10 (2010) 1857
[7] , Bridgeman's orthospectrum identity, Topology Proc. 38 (2011) 173
[8] , , , Complete flat Lorentz $3$–manifolds and laminations on hyperbolic surfaces, in preparation
[9] , editor, Structural properties of polylogarithms, Mathematical Surveys and Monographs 37, Amer. Math. Soc. (1991)
[10] , Kleinian groups, Grundl. Math. Wissen. 287, Springer (1988)
[11] , The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series 143, Cambridge Univ. Press (1989)
[12] , , Shortening all the simple closed geodesics on surfaces with boundary, Proc. Amer. Math. Soc. 138 (2010) 1775
[13] , Lengths of geodesics on Riemann surfaces with boundary, Ann. Acad. Sci. Fenn. Math. 30 (2005) 227
[14] , On function sum theorems connected with the series formula, Proc. London Math. Soc. S2-4 (1907) 169
[15] , The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. (1979) 171
Cité par Sources :