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# A note about stable transitivity of noncompact extensions of hyperbolic systems

• Let $f:X\to X$ be the restriction to a hyperbolic basic set of a smooth diffeomorphism. If $G$ is the special Euclidean group $SE(2)$ we show that in the set of $C^2$ $G$-extensions of $f$ there exists an open and dense subset of stably transitive transformations. If $G=K\times \mathbb R^n$, where $K$ is a compact connected Lie group, we show that an open and dense set of $C^2$ $G$-extensions satisfying a certain separation condition are transitive. The separation condition is necessary.
Mathematics Subject Classification: 37D20, 37D30.

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