Stable transitivity of certain noncompact extensions of hyperbolic systems

Annales Henri Poincaré 6 (2005) 725-746.

Ian Melbourne, Viorel Niţică and Andrei Török

Abstract

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 matrix 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. For the Euclidean groups SE(n) with n>2 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 x 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 generalises a result of Niţică and Pollicott for Rⁿ-extensions.