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April  2006, 14(2): 355-363. doi: 10.3934/dcds.2006.14.355

A note about stable transitivity of noncompact extensions of hyperbolic systems

Ian Melbourne 1, , V. Niţicâ 2, and  Andrei Török 3,

1. 

Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH
2. 

Department of Mathematics, West Chester University, West Chester, PA 19383, United States
3. 

Department of Mathematics, University of Houston, Houston, TX 77204-3008

Received  November 2004 Revised  February 2005 Published  November 2005

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.
Keywords: skew-product, stable transitivity, hyperbolic basic set., Noncompact group extension.
Mathematics Subject Classification: 37D20, 37D3.
Citation: Ian Melbourne, V. Niţicâ, Andrei Török. A note about stable transitivity of noncompact extensions of hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 355-363. doi: 10.3934/dcds.2006.14.355
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