Comptes Rendus
Dynamical Systems
Discretization of harmonic measures for foliated bundles
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 621-626.

We prove in this Note that there is, for some foliated bundles, a bijective correspondence between Garnettʼs harmonic measures and measures on the fiber that are stationary for some probability measure on the holonomy group. As a consequence, we show the uniqueness of the harmonic measure in the case of some foliations transverse to projective fiber bundles.

On prouve dans cette Note quʼil y a, pour certains fibrés feuilletés, une correspondance bijective entre les mesures harmoniques au sens de Garnett et les mesures sur la fibre qui sont stationnaires pour une certaine mesure de probabilité sur le groupe dʼholonomie. Nous en déduisons lʼunicité de la mesure harmonique pour certains feuilletages transverses à une fibration projective.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.06.010

Sébastien Alvarez 1

1 Institut de mathématiques de Bourgogne, CNRS-UMR 5584, Université de Bourgogne, 21078 Dijon cedex, France
@article{CRMATH_2012__350_11-12_621_0,
     author = {S\'ebastien Alvarez},
     title = {Discretization of harmonic measures for foliated bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {621--626},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.06.010},
     language = {en},
}
TY  - JOUR
AU  - Sébastien Alvarez
TI  - Discretization of harmonic measures for foliated bundles
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 621
EP  - 626
VL  - 350
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2012.06.010
LA  - en
ID  - CRMATH_2012__350_11-12_621_0
ER  - 
%0 Journal Article
%A Sébastien Alvarez
%T Discretization of harmonic measures for foliated bundles
%J Comptes Rendus. Mathématique
%D 2012
%P 621-626
%V 350
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2012.06.010
%G en
%F CRMATH_2012__350_11-12_621_0
Sébastien Alvarez. Discretization of harmonic measures for foliated bundles. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 621-626. doi : 10.1016/j.crma.2012.06.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.010/

[1] W. Ballmann; F. Ledrappier Discretization of positive harmonic functions on Riemannian manifolds and Martin boundary, Luminy, 1992 (Sémin. Congr.), Volume vol. 1, Soc. Math. Fr., Paris (1996), pp. 77-92

[2] C. Bonatti; X. Gómez-Mont Sur le comportement statistique des feuilles de certains feuilletages holomorphes, Monogr. Enseign. Math., vol. 38, 2001 (pp. 15–41)

[3] C. Bonatti; X. Gómez-Mont; M. Viana Généricité dʼexposants de Lyapunov non-nuls pour des produits déterministes de matrices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003), pp. 579-624

[4] C. Camacho; A. Lins Neto Geometric Theory of Foliations, Birkhäuser Boston Inc., 1985

[5] H. Furstenberg Noncommuting random products, Trans. Amer. Math. Soc., Volume 108 (1963), pp. 377-428

[6] H. Furstenberg Random walks and discrete subgroups of Lie groups, Adv. Probab. Relat. Top., vol. 1, Dekker, New York, 1971, pp. 1-63

[7] L. Garnett Foliations, the ergodic theorem and Brownian motion, J. Funct. Anal., Volume 51 (1983), pp. 285-311

[8] Y. Guivarcʼh; A. Raugi Frontière de Furstenberg, propriété de contraction et théorèmes de convergence, Z. Wahrscheinlichkeitstheor. Verw. Geb., Volume 69 (1985), pp. 187-242

[9] T. Lyons; D. Sullivan Function theory, random paths and covering spaces, J. Differential Geom., Volume 19 (1984), pp. 299-323

[10] V.A. Rokhlin On the fundamental ideas of measure theory, Trans. Amer. Math. Soc., Volume 10 (1962), pp. 1-52

Cited by Sources:

Comments - Policy