Generic measures for geodesic flows on nonpositively curved manifolds

Yves Coudène[1]; Barbara Schapira[2]

  • [1] Laboratoire de mathématiques, UBO 6 avenue le Gorgeu, 29238 Brest, France
  • [2] LAMFA, UMR CNRS 7352, Université Picardie Jules Verne 33 rue St Leu, 80000 Amiens, France

Journal de l’École polytechnique — Mathématiques (2014)

  • Volume: 1, page 387-408
  • ISSN: 2270-518X

Abstract

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We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result: ergodicity is a generic property in the space of all invariant measures defined on the unit tangent bundle of the surface if and only if there are no flat strips in the universal cover of the surface.Finally, we show under suitable assumptions that generically, the invariant probability measures have zero entropy and are not strongly mixing.

How to cite

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Coudène, Yves, and Schapira, Barbara. "Generic measures for geodesic flows on nonpositively curved manifolds." Journal de l’École polytechnique — Mathématiques 1 (2014): 387-408. <http://eudml.org/doc/275502>.

@article{Coudène2014,
abstract = {We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result: ergodicity is a generic property in the space of all invariant measures defined on the unit tangent bundle of the surface if and only if there are no flat strips in the universal cover of the surface.Finally, we show under suitable assumptions that generically, the invariant probability measures have zero entropy and are not strongly mixing.},
affiliation = {Laboratoire de mathématiques, UBO 6 avenue le Gorgeu, 29238 Brest, France; LAMFA, UMR CNRS 7352, Université Picardie Jules Verne 33 rue St Leu, 80000 Amiens, France},
author = {Coudène, Yves, Schapira, Barbara},
journal = {Journal de l’École polytechnique — Mathématiques},
keywords = {Geodesic flow; hyperbolic dynamical systems; nonpositive curvature; ergodicity; generic measures; zero entropy; mixing; geodesic flow},
language = {eng},
pages = {387-408},
publisher = {École polytechnique},
title = {Generic measures for geodesic flows on nonpositively curved manifolds},
url = {http://eudml.org/doc/275502},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Coudène, Yves
AU - Schapira, Barbara
TI - Generic measures for geodesic flows on nonpositively curved manifolds
JO - Journal de l’École polytechnique — Mathématiques
PY - 2014
PB - École polytechnique
VL - 1
SP - 387
EP - 408
AB - We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set.In the case of a compact surface, we get the following sharp result: ergodicity is a generic property in the space of all invariant measures defined on the unit tangent bundle of the surface if and only if there are no flat strips in the universal cover of the surface.Finally, we show under suitable assumptions that generically, the invariant probability measures have zero entropy and are not strongly mixing.
LA - eng
KW - Geodesic flow; hyperbolic dynamical systems; nonpositive curvature; ergodicity; generic measures; zero entropy; mixing; geodesic flow
UR - http://eudml.org/doc/275502
ER -

References

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