Skip to main content Accessibility help

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



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.



Hide All
[Boh11]Bohnet, D.. Partially hyperbolic diffeomorphisms with a compact center foliation with finite holonomy. PhD Thesis, University of Hamburg, 2011.
[Boh13]Bohnet, D.. Codimension one partially hyperbolic diffeomorphisms with uniformly compact center foliation. J. Mod. Dyn. 7(4) (2013), 140.
[Bon93]Bonatti, C.. Feuilletages proches d’une fibration (Ensaios matemáticos, 5). Sociedade Brasileira de Matemática, Rio de Janeiro, Brazil, 1993.
[Bri03]Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23(2) (2003), 395401.
[BW05]Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475508.
[Car10]Carrasco, P.. Compact dynamical foliations. PhD Thesis, University of Toronto, 2010.
[Car11]Carrasco, P.. Compact dynamical foliations. Ergod. Th. & Dynam. Sys. (2011), to appear, Preprint,arXiv:1105.0052v2.
[CC00]Candel, A. and Conlon, L.. Foliations. I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2000.
[EMT77]Epstein, D. B. A., Millet, K. and Tischler, D.. Leaves without holonomy. J. Lond. Math. Soc. (2) 16(3) (1977), 548552.
[Eps76]Epstein, D. B. A.. Foliations with all leaves compact. Ann. Inst. Fourier (Grenoble) 26(1) (1976), 265282.
[EV78]Epstein, D. B. A. and Vogt, E.. A counterexample to the periodic orbit conjecture in codimension 3. Ann. of Math. (2) 108(3) (1978), 539552.
[Gog11]Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Mod. Dyn. 5 (2011), 747767, arXiv:1104.5464v2.
[Gog12]Gogolev, A.. How typical are pathological foliations in partially hyperbolic dynamics: an example. Israel J. Math. 187 (2012), 493507.
[Gou07]Gourmelon, N.. Adapted metric for dominated splittings. Ergod. Th. & Dynam. Sys. 27(6) (2007), 18391849.
[Hec77]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. 252270.
[HPS70]Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant manifolds. Bull. Amer. Math. Soc. 76 (1970), 10151019.
[HRHU10]Rodriguez Hertz, F., Rodriguez-Hertz, M. A. and Ures, R.. A non-dynamically coherent example in 3-torus. Preprint, 2010.
[KT12]Kryzhevich, S. and Tikhomirov, S.. Partial hyperbolicity and central shadowing. Discrete Contin. Dyn. Syst. 33(7) (2013), 29012909.
[Lew89]Lewowicz, J.. Expansive homeomorphisms of surfaces. Bol. Soc. Brasil. Math. (NS) 20(1) (1989), 113133.
[MM03]Moerdijk, I. and Mrčun, J.. Introduction to Foliations and Lie Groupoids. Cambridge University Press, Cambridge, UK, 2003.
[Sul76]Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. (46) (1976), 514.
[SW00]Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139(3) (2000), 495508.
[Wil98]Wilkinson, A.. Stable ergodicity of the time-one map of a geodesic flow. Ergod. Th. & Dynam. Sys. 18(6) (1998), 15451587.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed