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

Axioms

Viorel Nitica¹  Andrei Török² 

[1] Department of Mathematics, West Chester University, West Chester, PA 19383, USA;Institute of Mathematics of the Romanian Academy, P.O. Box 1–764, RO-70700 Bucharest, Romania

关键词: stable topological transitivity;    hyperbolic system;    Anosov diffeomorphism;    extension with Lie group fiber;   

DOI  :  10.3390/axioms4010084

来源: mdpi

【 摘 要 】

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.

【 授权许可】

CC BY   
© 2015 by the authors; licensee MDPI, Basel, Switzerland.