Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T14:37:46.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Partially hyperbolic diffeomorphisms with a uniformly compact center foliation: the quotient dynamics

Published online by Cambridge University Press:  11 February 2015

DORIS BOHNET  and
CHRISTIAN BONATTI
DORIS BOHNET
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Dijon, France email [email protected], [email protected]
CHRISTIAN BONATTI
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584 CNRS, Université de Bourgogne, Dijon, France email [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 4 , June 2016 , pp. 1067 - 1105
Copyright
© Cambridge University Press, 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bohnet, D.. Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy. PhD Thesis, University of Hamburg, 2011.Google Scholar
Bohnet, D.. Codimension one partially hyperbolic diffeomorphisms with uniformly compact center foliation. J. Mod. Dyn. 7(4) (2013), 1–40.Google Scholar
Bonatti, C.. Feuilletages proches d’une fibration (Ensaios matemáticos, 5). Sociedade Brasileira de Matemática, Rio de Janeiro, Brazil, 1993.Google Scholar
Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23(2) (2003), 395–401.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475–508.Google Scholar
Carrasco, P.. Compact dynamical foliations. PhD Thesis, University of Toronto, 2010.Google Scholar
Carrasco, P.. Compact dynamical foliations. Ergod. Th. & Dynam. Sys. (2011), to appear, Preprint,arXiv:1105.0052v2.Google Scholar
Candel, A. and Conlon, L.. Foliations. I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2000.Google Scholar
Epstein, D. B. A., Millet, K. and Tischler, D.. Leaves without holonomy. J. Lond. Math. Soc. (2) 16(3) (1977), 548–552.Google Scholar
Epstein, D. B. A.. Foliations with all leaves compact. Ann. Inst. Fourier (Grenoble) 26(1) (1976), 265–282.Google Scholar
Epstein, D. B. A. and Vogt, E.. A counterexample to the periodic orbit conjecture in codimension 3. Ann. of Math. (2) 108(3) (1978), 539–552.CrossRefGoogle Scholar
Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Mod. Dyn. 5 (2011), 747–767, arXiv:1104.5464v2.Google Scholar
Gogolev, A.. How typical are pathological foliations in partially hyperbolic dynamics: an example. Israel J. Math. 187 (2012), 493–507.Google Scholar
Gourmelon, N.. Adapted metric for dominated splittings. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1839–1849.Google Scholar
Hector, G.. Feuilletages en cylindres. Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) (Lecture Notes in Mathematics, 597). Springer, Berlin, 1977, pp. 252–270.Google Scholar
Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant manifolds. Bull. Amer. Math. Soc. 76 (1970), 1015–1019.Google Scholar
Rodriguez Hertz, F., Rodriguez-Hertz, M. A. and Ures, R.. A non-dynamically coherent example in 3-torus. Preprint, 2010.Google Scholar
Kryzhevich, S. and Tikhomirov, S.. Partial hyperbolicity and central shadowing. Discrete Contin. Dyn. Syst. 33(7) (2013), 2901–2909.Google Scholar
Lewowicz, J.. Expansive homeomorphisms of surfaces. Bol. Soc. Brasil. Math. (NS) 20(1) (1989), 113–133.Google Scholar
Moerdijk, I. and Mrčun, J.. Introduction to Foliations and Lie Groupoids. Cambridge University Press, Cambridge, UK, 2003.Google Scholar
Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. (46) (1976), 5–14.Google Scholar
Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139(3) (2000), 495–508.Google Scholar
Wilkinson, A.. Stable ergodicity of the time-one map of a geodesic flow. Ergod. Th. & Dynam. Sys. 18(6) (1998), 1545–1587.Google Scholar