×

zbMATH — the first resource for mathematics

Partial hyperbolicity for symplectic diffeomorphisms. (English) Zbl 1130.37356
Summary: We prove that every robustly transitive and every stably ergodic symplectic diffeomorphism on a compact manifold admits a dominated splitting. In fact, these diffeomorphisms are partially hyperbolic.

MSC:
37D30 Partially hyperbolic systems and dominated splittings
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
References:
[1] A. Arbieto, C. Matheus, A pasting lemma I: the case of vector fields, Preprint, IMPA, 2003
[2] Arnaud, M.-C., The generic symplectic \(C^1\)-diffeomorphisms of four-dimensional symplectic manifolds are hyperbolic, partially hyperbolic or have a completely elliptic periodic point, Ergodic theory dynam. systems, 22, 6, 1621-1639, (2002) · Zbl 1030.37037
[3] J. Bochi, M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, Preprint IMPA, 2003 · Zbl 1147.37315
[4] Bonatti, C.; Díaz, L.J., Nonhyperbolic transitive diffeomorphisms, Ann. of math., 143, 357-396, (1996) · Zbl 0852.58066
[5] Bonatti, C.; Díaz, L.J.; Pujals, E., A \(C^1\)-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources, Ann. of math., 157, 2, 355-418, (2003) · Zbl 1049.37011
[6] Bonatti, C.; Viana, M., SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. math., 115, 157-193, (2000) · Zbl 0996.37033
[7] Burns, K.; Pugh, C.; Shub, M.; Wilkinson, A., Recent results about stable ergodicity, Proc. sympos. amer. math. soc., 69, 327-366, (2001) · Zbl 1012.37019
[8] Dacorogna, B.; Moser, J., On a partial differential equation involving the Jacobian determinant, Ann. inst. H. Poincaré anal. non linéaire, 7, 1, 1-26, (1990) · Zbl 0707.35041
[9] Díaz, L.J.; Pujals, E.; Ures, R., Partial hyperbolicity and robust transitivity, Acta math., 183, 1-43, (1999) · Zbl 0987.37020
[10] Mañé, R., Contributions to the stability conjecture, Topology, 17, 383-396, (1978) · Zbl 0405.58035
[11] Mañé, R., An ergodic closing lemma, Ann. of math., 116, 503-540, (1982) · Zbl 0511.58029
[12] Mañé, R., Oseledec’s theorem from the generic viewpoint, (), 1269-1276
[13] Newhouse, S., Quasi-elliptic periodic points in conservative dynamical systems, Amer. J. math., 99, 5, 1061-1087, (1976) · Zbl 0379.58011
[14] Shub, M., Topologically transitive diffeomorphisms on \(T^4\), (), 39
[15] Tahzibi, A., Stably ergodic systems which are not partially hyperbolic, Israel J. math., 24, 204, 315-342, (2004) · Zbl 1052.37019
[16] Vivier, T., Flots robustament transitif sur des variété compactes, C. R. math. acad. sci. Paris, 337, 12, 791-796, (2003) · Zbl 1079.37013
[17] Xia, Z., Homoclinic points in symplectic and volume-preserving diffeomorphisms, Comm. math. phys., 177, 2, 435-449, (1996) · Zbl 0959.37050
[18] Zehnder, E., A note on smoothing symplectic and volume preserving diffeomorphisms, (), 828-854
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.