# Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

## Authors

- First Online:

DOI: 10.1007/s00023-005-0221-0

- Cite this article as:
- Melbourne, I., Niţică, V. & Török, A. Ann. Henri Poincaré (2005) 6: 725. doi:10.1007/s00023-005-0221-0

- 5 Citations
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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