Annales Henri Poincaré

, Volume 6, Issue 4, pp 725–746

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Original Paper

DOI: 10.1007/s00023-005-0221-0

Cite this article as:
Melbourne, I., Niţică, V. & Török, A. Ann. Henri Poincaré (2005) 6: 725. doi:10.1007/s00023-005-0221-0

Abstract.

Let \(f:X \to X\) be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finite-dimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive \(SL(2,\mathbb{R})\)-extensions. More generally, we find stably transitive \(Sp(2n,\mathbb{R})\)-extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive.

For groups of the form \(K \times \mathbb{R}^n \) where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalizes a result of Niţică and Pollicott for \(\mathbb{R}^n\) -extensions.

Communicated by Viviane Baladi

Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  • Ian Melbourne
    • 1
  • Viorel Niţică
    • 2
    • 3
  • Andrei Török
    • 4
    • 3
  1. 1.Department of Mathematics and StatisticsUniversity of SurreyGuildfordUnited Kingdom
  2. 2.Department of MathematicsWest Chester UniversityWest ChesterUSA
  3. 3.Institute of Mathematics of the Romanian AcademyBucharestRomania
  4. 4.Department of MathematicsUniversity of HoustonHoustonUSA

Personalised recommendations