Annales Henri Poincaré

, Volume 6, Issue 4, pp 725–746 | Cite as

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Original Paper


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


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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

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