Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-02-05T14:01:31.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact dynamical foliations

Published online by Cambridge University Press:  04 August 2014

PABLO D. CARRASCO
PABLO D. CARRASCO*
Affiliation:
I.M.P.A., Estrada Dona Castorina 110 CEP 22360 420, Rio de Janeiro, Brazil email pdcarrasco@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

According to the work of Dennis Sullivan [A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14], there exists a smooth flow on the five-sphere all of whose orbits are periodic, although there is no uniform bound on their periods. The question addressed in this article is whether these type of examples can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact but there is no uniform bound for their volumes? We develop tools to attack the previous question and show that it has a negative answer provided that all periodic leaves have finite holonomy.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 8 , December 2015 , pp. 2474 - 2498
Copyright
© Cambridge University Press, 2014 

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

Adler, R., Kitchens, B. and Shub, M.. Stably ergodic skew products. Discrete Contin. Dyn. Syst. 2 (1996), 349–350.CrossRefGoogle Scholar
Anosov, D.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Math. Inst. 90 (1967), 1–235.Google Scholar
Berger, P.. Persistence of stratification of normally expanded laminations. Mém. Soc. Math. Fr. (N.S.) 134 (2013), 1–133.Google Scholar
Bohnet, D.. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. J. Modern Dyn. 4 (2013), 565–604.CrossRefGoogle Scholar
Bonatti, C. and Bohnet, D.. Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics. Preprint, 2012, http://arxiv.org/abs/1210.2835.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475–508.CrossRefGoogle Scholar
Bourbaki, N.. Elements of General Topology: Part 2. Springer, Berlin, 1998, reprint of the 1989 English translation edition.Google Scholar
Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.CrossRefGoogle Scholar
Brin, M.. The topology of group extensions of C systems. Math. Notes 18 (1975), 858–864.CrossRefGoogle Scholar
Brin, M.. On dynamical coherence. Ergod. Th.& Dynam. Sys. 23(2) (2003), 395–401.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. Stable ergodicity of skew products. Ann. Sci. Éc. Norm. Supér. (4) 32(6) (1999), 859–889.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. Dynamical coherence, accessibility and center bunching. Discrete Contin. Dyn. Syst. 22 (2008), 89–100 (Pesin birthday issue).Google Scholar
Carrasco, P. D.. Compact dynamical foliations. PhD Thesis, University of Toronto, 2011.Google Scholar
Candel, A. and Conlon, L.. Foliations I (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2000.Google Scholar
Edwards, R., Millett, K. and Sullivan, D.. Foliations with all leaves compact. Topology 16(1) (1977), 13–32.CrossRefGoogle Scholar
Ehresmann, C. and Shih, W.. Sur les espaces feuilltees: theoreme de stabilite. C. R. Math. Acad. Sci. Paris 243 (1956), 344–346.Google Scholar
Epstein, D. B. A.. Periodic flows on three-manifolds. Ann. of Math. (2) 5(95) (1972), 66–82.CrossRefGoogle Scholar
Epstein, D. B. A.. Foliations with all leaves compact. Ann. Inst. Fourier (Grenoble) 26(1) (1976), 265–282.CrossRefGoogle Scholar
Epstein, D. B. A.. Pointwise periodic homeomorphisms. Proc. Lond. Math. Soc. s3-42(3) (1981), 415–460.CrossRefGoogle Scholar
Epstein, D. B. A., Millett, K. and Tischler, D.. Leaves without holonomy. J. Lond. Math. Soc. (2) 16(3) (1977), 548–552.CrossRefGoogle 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
Field, M. and Parry, W.. Stable ergodicity of skew extensions by Lie groups. Topology 38(1) (1999), 167–187.CrossRefGoogle Scholar
Gogolev, A.. Partially hyperbolic diffeomorphisms with compact center foliations. J. Modern Dyn. 5(4) (2011), 747–767.CrossRefGoogle Scholar
Gourmelon, N.. Adapted metrics for dominated splittings. Ergod. Th. & Dynam. Sys. 27 (2007), 1839–1849.CrossRefGoogle Scholar
Haefliger, A.. Varietes feuilletees. Ann. Sc. Norm. Super. Pisa XVI(IV) (1962), 367–397.Google Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
Kelley, J. L.. General Topology (Graduate Text in Mathematics, 27). Springer, New York, 1975, reprint of the 1955 edition.Google Scholar
Mané, R.. Introdução à teoria ergódica (Projecto Euclides). IMPA, Brazil, 1979.Google Scholar
Montgomery, D.. Pointwise periodic homeomorphisms. Amer. J. Math. 59(1) (1937), 118–120.CrossRefGoogle Scholar
Pesin, Y.. Lectures on Partial Hyperbolicity and Stable Ergodicity (Zurich Lectures in Advanced Mathematics). European Mathematical Society, 2004.CrossRefGoogle Scholar
Pugh, C. and Shub, M.. Ergodicity of Anosov actions. Invent. Math. 15 (1972), 1–23.CrossRefGoogle Scholar
Pugh, C. and Shub, M.. Stably ergodic dynamical systems and partial hyperbolicity. J. Complexity 13 (1997), 125–179.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations. Duke Math. J. 86(3) (1997), 517–546.CrossRefGoogle Scholar
Pugh, C., Shub, M. and Wilkinson, A.. Hölder foliations, revisited. J. Modern Dyn. 6(1) (2012), 79–120.CrossRefGoogle Scholar
Pugh, C.. Private communication, March 2011.Google Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, M. and Ures, R.. A survey of partially hyperbolic dynamics. Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow (Fields Institute Communications, 51). Eds. Forni, G., Lyubich, M., Pugh, C. and Shub, M.. Fields Institute, Toronto, 2007, pp. 35–88.Google Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, M. and Ures, R.. On the existence and uniqueness of weak foliations in dimension 3. Geom. Probab. Struct. Dynam. 469 (2008), 303–316.CrossRefGoogle Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, M. and Ures, R.. A non-dynamical coherent example in $\mathbb{T}^{3}$. 2010, in preparation.Google Scholar
Smale, S.. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747–817.CrossRefGoogle Scholar
Sullivan, D.. A counterexample to the periodic orbit conjecture. Publ. Math. Inst. Hautes Études Sci. 46 (1976), 5–14.CrossRefGoogle Scholar
Shub, M. and Wilkinson, A.. Pathological foliations and removable zero exponents. Invent. Math. 139 (2000), 495–508.CrossRefGoogle Scholar
Vogt, E.. A periodic flow with infinite Epstein hierarchy. Manuscripta Math. 22(4) (1977), 403–412.CrossRefGoogle Scholar
Vogt, E.. Bad Sets of Compact Foliations of Codimension Two (Lectures in Low-dimensional Topology, 3). University of Tennessee, International Press Company, Knoxville, 1994, pp. 187–216.Google Scholar