Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-31T11:47:35.421Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrability of dominated decompositions on three-dimensional manifolds

Published online by Cambridge University Press:  11 February 2016

STEFANO LUZZATTO ,
SİNA TÜRELİ  and
KHADIM WAR
STEFANO LUZZATTO
Affiliation:
Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, Italy email luzzatto@ictp.it, sinatureli@gmail.com, kwar@ictp.it
SİNA TÜRELİ
Affiliation:
Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, Italy email luzzatto@ictp.it, sinatureli@gmail.com, kwar@ictp.it International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste, Italy
KHADIM WAR
Affiliation:
Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste, Italy email luzzatto@ictp.it, sinatureli@gmail.com, kwar@ictp.it International School for Advanced Studies (SISSA), Via Bonomea 265, Trieste, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 606 - 620
Copyright
© Cambridge University Press, 2016 

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.)

Article purchase

Temporarily unavailable

References

Agrachev, A. A. and Sachkov, Y. L.. Control Theory from the Geometric Viewpoint. Springer, Berlin, 2004.Google Scholar
Bonatti, C., Díaz, L. and Pujals, E.. A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Ann. of Math. 2(158) (2003), 355–418.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifold. Topology 44(3) (2005), 475–508.Google Scholar
Brin, M.. On dynamical coherence. Ergod. Th. & Dynam. Sys. 23(2) (2003), 395–401.Google Scholar
Brin, M. and Pesin, Y.. Flows of frames on manifolds of negative curvature. Uspekhi Mat. Nauk 28(4(172)) (1973), 209–210.Google Scholar
Brin, M. and Pesin, Y.. Partially hyperbolic dynamical systems. Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 170–212.Google Scholar
Brin, M., Burago, D. and Ivanov, S.. On Partially Hyperbolic Diffeomorphisms of 3-Manifolds with Commutative Fundamental Group (Modern Dynamical Systems and Applications) . Cambridge University Press, Cambridge, 2004, pp. 307–312.Google Scholar
Burago, D. and Ivanov, S.. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2(4) (2008), 541–580.Google Scholar
Burns, K. and Wilkinson, A.. Dynamical coherence and center bunching. Discrete Conti. Dyn. Syst. 22 (2008), 89–100.Google Scholar
Crovisier, S.. Partial hyperbolicity far from homoclinic tangencies. Adv. Math. 226 (2011), 673–726.Google Scholar
Díaz, L. D., Pujals, E. R. and Ures, R.. Partially hyperbolicity and robust transitivity. Acta Math. 183(1) (1999), 1–43.Google Scholar
Flavio, A.. Attractors of generic diffeomorphisms are persistent. Nonlinearity 16 (2003), 301–311.Google Scholar
Flavio, A., Bonatti, C. and Díaz, L.. Non-wondering sets with non-empty interiors. Nonlinearity 17 (2004), 175–191.Google Scholar
Hammerlindl, A.. Integrability and Lyapunov exponents. J. Mod. Dyn. 5(1) (2011), 107–122.CrossRefGoogle Scholar
Hartman, P.. Ordinary Differential Equations. Wiley, New York, 1964.Google Scholar
Hirsch, M., Pugh, M. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583) . Springer, Berlin, 1977.Google Scholar
Lee, J. M.. Introduction to Smooth Manifolds. Springer, New York, 2003.CrossRefGoogle Scholar
Luzzatto, S., Türeli, S. and War, K.. Integrability of $C^{1}$ invariant splittings. http://front.math.ucdavis.edu/1408.6948.Google Scholar
Luzzatto, S., Türeli, S. and War, K.. Integrability of dominated decompositions on three-dimensional manifolds (version 2). http://front.math.ucdavis.edu/1410.8072v2.Google Scholar
Luzzatto, S., Türeli, S. and War, K.. A Frobenius theorem in dimension 3. http://front.math.ucdavis.edu/1411.5896.Google Scholar
Mañé, R.. Contributions to the stability conjecture. Topology 17 (1978), 383–396.CrossRefGoogle Scholar
Parwani, K.. On 3-manifolds that support partially hyperbolic diffeomorphisms. Nonlinearity 23 (2010), 589–606.Google Scholar
Pesin, Y.. Lectures on Partial Hyperbolicity and Stable Ergodicity. European Mathematical Society, Zurich, 2004.CrossRefGoogle Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. A non-dynamically coherent example on $\mathbb{T}^{3}$ . http://front.math.ucdavis.edu/1409.0738.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. A survey of partially hyperbolic dynamics. Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow. American Mathematical Society, Providence, RI, 2007, pp. 103–112.Google Scholar
Rodriguez Hertz, F., Rodriguez Hertz, M. A. and Ures, R.. On existence and uniqueness of weak foliations in dimension 3. Contemp. Math. 469 (2008), 303–316.Google Scholar
Sambarion, M. and Potrie, R.. Codimension 1 generic homoclinic classes with interior. Bull. Braz. Math. Soc. (N.S.) 41(1) (2010), 125–138.Google Scholar
Simić, S.. Lipschitz distributions and Anosov flows. Proc. Amer. Math. Soc. 124(6) (1996), 1869–1877.Google Scholar
Türeli, S. and War, K.. Integrability for dominated splitting with linear growth in dimension 3. Preprint.Google Scholar
Viana, M. and Bonatti, Christian. SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115 (2000), 157–193.Google Scholar