Stable transitivity of certain noncompact extensions of hyperbolic systems

Abstract

<p>Let ƒ : <i>X</i> → <i>X</i> 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 <i>stably</i> transitive extensions. In particular, we find stably transitive <i>SL</i>(2, <b>R</b>)-extensions. More generally, we find stably transitive <i>Sp</i>(2<i>n</i>-<b>R)</b>-extensions for all <i>n</i> ≥ 1. For the Euclidean groups <i>SE</i>(<i>n</i>) with <i>n</i> ≥ 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. </p> <p>For groups of the form <i>K</i> × <b>R</b>^<i>n</i> where <i>K</i> is compact, a separation condition is necessary for transitivity. Provided <i>X</i> 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 Nitica and Pollicott for <b>R</b>^<i>n</i>-extensions.</p>

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