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Gibbs u-states for the foliated geodesic flow and transverse invariant measures
- Source :
- Israel Journal of Mathematics. 221:869-940
- Publication Year :
- 2017
- Publisher :
- Springer Science and Business Media LLC, 2017.
-
Abstract
- This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation with negatively curved leaves. On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when this foliated geodesic flow preserves a Gibbs su-state, i.e. a measure with Lebesgue disintegration both in the stable and unstable horospheres, then it has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. On the other hand we study in detail the projections of Gibbs u-states along the unit spheres tangent to the foliation. We show that they have Lebesgue disintegration in the leaves and that the local densities possess an integral representation analogue to the Poisson representation of harmonic functions.<br />Comment: 50 pages, 13 figures, to appear in Israel Journal of Mathematics
- Subjects :
- Pure mathematics
Mathematics::Dynamical Systems
General Mathematics
010102 general mathematics
Dynamical Systems (math.DS)
Lebesgue integration
01 natural sciences
Manifold
symbols.namesake
Harmonic function
Unit tangent bundle
0103 physical sciences
FOS: Mathematics
Foliation (geology)
symbols
Mathematics::Differential Geometry
010307 mathematical physics
Invariant measure
Mathematics - Dynamical Systems
0101 mathematics
Invariant (mathematics)
Probability measure
Mathematics
Subjects
Details
- ISSN :
- 15658511 and 00212172
- Volume :
- 221
- Database :
- OpenAIRE
- Journal :
- Israel Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....a695fbda05b42cc3ce0ffa8cd3790b37