Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Let f : X → 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 finitedimensional 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,R)extensions. More generally, we find stably transitive Sp(2n,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 × Rn 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 Rn-extensions.