Original Paper

Annales Henri Poincaré

, Volume 6, Issue 4, pp 725-746

First online:

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

  • Ian MelbourneAffiliated withDepartment of Mathematics and Statistics, University of Surrey Email author 
  • , Viorel NiţicăAffiliated withDepartment of Mathematics, West Chester UniversityInstitute of Mathematics of the Romanian Academy
  • , Andrei TörökAffiliated withDepartment of Mathematics, University of HoustonInstitute of Mathematics of the Romanian Academy

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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