A note about stable transitivity of noncompact extensions of hyperbolic systems

Ian Melbourne; V. Niţicâ; Andrei Török

doi:10.3934/dcds.2006.14.355

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2006, Volume 14, Issue 2: 355-363. Doi: 10.3934/dcds.2006.14.355

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A note about stable transitivity of noncompact extensions of hyperbolic systems

1.
Department of Mathematics and Statistics, University of Surrey, Guildford, Surrey GU2 7XH
2.
Department of Mathematics, West Chester University, West Chester, PA 19383
3.
Department of Mathematics, University of Houston, Houston, TX 77204-3008

Received: November 2004

Revised: February 2005

Published: April 2006

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Abstract

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.

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Mathematics Subject Classification: 37D20, 37D30.

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