Chaotic extensions of dynamical systems by function algebras
A topological dynamical system is by definition a compact Hausdorff space X with an action by the semigroup \({\mathbb{N}}\) or the group \({\mathbb{Z}}\) by continuous surjective maps. In terms of the \(C^*\)-algebra C(X) this means there is a *-isometry \(\phi\) on C(X). Every \(C^*\)-extension of C(X) and \(\phi\) will give an extension of the dynamical system. A \(\phi\)- stable \(C^*\)-extension A lying inside the algebra B(X) of bounded functions on X is called tame if A is generated by functions with finitely many points of discontinuity and A does not contain characteristic functions of singletons. The principal question of the paper is which topological properties of dynamical systems are preserved under tame extensions. We learn that transitivity, mixing, denseness of periodic points and chaos are preserved under tame extensions. On the other hand examples are given to show that expansiveness and topological entropy are not always preserved. The main tool is an explicit construction of the space of a tame extension in terms of the generators of A and the original space X.
scientific article; zbMATH DE number 4214926
scientific article