Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-17T21:33:29.015Z Has data issue: false hasContentIssue false
  • English
  • Français

CENTER MANIFOLDS FOR PARTIALLY HYPERBOLIC SETS WITHOUT STRONG UNSTABLE CONNECTIONS

Part of: Smooth dynamical systems: general theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  11 March 2015

Christian Bonatti  and
Sylvain Crovisier
Christian Bonatti
Affiliation:
Institut de Mathématiques de Bourgogne, CNRS - URM 5584, Université de Bourgogne, Dijon 21004, France (bonatti@u-bourgogne.fr)
Sylvain Crovisier
Affiliation:
Laboratoire de Mathématiques d’Orsay, CNRS - UMR 8628, Université Paris-Sud 11, Orsay 91405, France (Sylvain.Crovisier@math.u-psud.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set $K$ is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects $K$ at exactly one point.

Keywords

center manifoldpartial hyperbolicityheteroclinic intersection

MSC classification

Primary: 37D10: Invariant manifold theory
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37C29: Homoclinic and heteroclinic orbits
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 4 , October 2016 , pp. 785 - 828
Copyright
© Cambridge University Press 2015 

References

Abraham, R. and Robbin, J., Transversal mappings and flows (W. A. Benjamin, Inc., New York-Amsterdam, 1967).Google Scholar
Bonatti, C., Díaz, L. and Viana, M., Dynamics Beyond Uniform Hyperbolicity (Springer, Berlin, 2004).Google Scholar
Chow, S.-N., Liu, W., Yi, Y. and Yingfei, Center manifolds for invariant sets, J. Differential Equations 168 (2000), 355–385.CrossRefGoogle Scholar
Crovisier, S., Periodic orbits and chain-transitive sets of C 1 -diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 104 (2006), 87–141.CrossRefGoogle Scholar
Crovisier, S., Partial hyperbolicity far from homoclinic bifurcations, Adv. Math. 226 (2011), 673–726.Google Scholar
Crovisier, S. and Gourmelon, N., Stabilisation of homoclinic tangencies in higher dimension. In preparation.Google Scholar
Crovisier, S. and Pujals, E. R., Essential hyperbolicity versus homoclinic bifurcations. Invent. Math. to appear.Google Scholar
Crovisier, S., Pujals, E. R. and Sambarino, M., Hyperbolicity of extremal bundles. In preparation.Google Scholar
de Melo, W., Structural stability of diffeomorphisms on two-manifolds, Invent. Math. 21 (1973), 233–246.Google Scholar
Gourmelon, N., Adapted metrics for dominated splittings, Ergodic Theory Dynam. Systems 27 (2007), 1839–1849.Google Scholar
Gourmelon, N., Generation of homoclinic tangencies by C 1 -perturbations, Discrete Contin. Dyn. Syst. 26 (2010), 1–42.Google Scholar
Hirsch, M., Pugh, C. and Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, volume 583 (Springer-Verlag, Berlin, 1977).Google Scholar
Mañé, R., Hyperbolicity, sinks and measure in one dimensional dynamics, Commun. Math. Phys. 100 (1985), 495–524. and Commun. Math. Phys. 112 (1987), 721–724.CrossRefGoogle Scholar
Newhouse, S., Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18.Google Scholar
Newhouse, S., The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. Inst. Hautes Études Sci. 50 (1979), 101–151.Google Scholar
Palis, J. and Takens, F., Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35 (Cambridge University Press, 1993).Google Scholar
Palis, J. and Viana, M., High dimension diffeomorphisms displaying infinitely many periodic attractors, Ann. of Math. 140(1) (1994), 207–250.Google Scholar
Pujals, E. R., On the density of hyperbolicity and homoclinic bifurcations for 3D- diffeomorphisms in attracting regions, Discrete Contin. Dyn. Syst. 16 (2006), 179–226.Google Scholar
Pujals, E. R. and Sambarino, M., Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), 961–1023.CrossRefGoogle Scholar
Pujals, E. R. and Sambarino, M., Density of hyperbolicity and tangencies in sectional dissipative regions, Ann. Inst. H. Poincaré 26 (2009), 1971–2000.Google Scholar
Romero, N., Persistence of homoclinic tangencies in higher dimensions, Ergodic Theory Dynam. Systems 15 (1995), 735–757.CrossRefGoogle Scholar
Wen, L., Generic Diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. 35 (2004), 419–452.Google Scholar
Wiggins, S., Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Applied Mathematical Sciences, vol. 105 (Springer-Verlag, 1994).Google Scholar