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Axioms 2015, 4(1), 84-101; doi:10.3390/axioms4010084

Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review

1
Department of Mathematics, West Chester University, West Chester, PA 19383, USA
2
Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania
3
Department of Mathematics, University of Houston, 651 PGH, Houston, TX 77204-3008, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Sidney A. Morris
Received: 12 December 2014 / Revised: 9 January 2015 / Accepted: 26 January 2015 / Published: 4 February 2015
(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)
View Full-Text   |   Download PDF [267 KB, uploaded 4 February 2015]

Abstract

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics. View Full-Text
Keywords: stable topological transitivity; hyperbolic system; Anosov diffeomorphism; extension with Lie group fiber stable topological transitivity; hyperbolic system; Anosov diffeomorphism; extension with Lie group fiber
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Nitica, V.; Török, A. Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review. Axioms 2015, 4, 84-101.

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