, Volume 6, Issue 4, pp 725746
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 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 \(SL(2,\mathbb{R})\)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
 Title
 Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems
 Journal

Annales Henri Poincaré
Volume 6, Issue 4 , pp 725746
 Cover Date
 200508
 DOI
 10.1007/s0002300502210
 Print ISSN
 14240637
 Online ISSN
 14240661
 Publisher
 BirkhäuserVerlag
 Additional Links
 Topics
 Authors

 Ian Melbourne ^{(1)}
 Viorel Niţică ^{(2)} ^{(3)}
 Andrei Török ^{(3)} ^{(4)}
 Author Affiliations

 1. Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom
 2. Department of Mathematics, West Chester University, 323 Anderson Hall, West Chester, PA, 19383, USA
 3. Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO70700, Bucharest, Romania
 4. Department of Mathematics, University of Houston, 651 PGH, Houston, TX, 772043008, USA