, Volume 6, Issue 4, pp 725-746

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

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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

submitted 24/05/04, accepted 11/10/04