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.
Highlights
A dynamical system is a continuous map f of a topological space X
The original definition of topological chaos given by Devaney, in addition to topological transitivity, requires the existence of a dense set of periodic points and the sensitive dependence of initial data for the dynamical system
Is compact topological space and the continuous map f has an invariant Borel probability measure μ, which is positive on open sets, topological transitivity is implied by ergodicity
Summary
A dynamical system is a continuous map f of a topological space X. If X is compact topological space and the continuous map f has an invariant Borel probability measure μ, which is positive on open sets, topological transitivity is implied by ergodicity. Brin [23] proved that a C 1 diffeomorphism that has an accessible pair of stable/unstable foliations and a dense set of recurrent points is topologically transitive. A robust obstruction to topological transitivity is the existence of a trapping region, i.e., a non-empty open proper subset U ⊂ M , such that f (Ū ) ⊆ U When this obstruction does not occur, it follows from the work of Bonatti and Crovisier [26] that a generic C 1 diffeomorphism of a compact Riemannian manifold is topologically transitive. These classes of dynamical systems can be thought of as thin classes of partially hyperbolic systems
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