Topological Entropy and Mixing Invariant Extremal Distributional Chaos
We show that there exists a mixing dynamical system with an invariant, extremal and transitive distributionally scrambled set. Meanwhile, we prove that such a complex dynamical system has zero topological entropy. Finally, we extend the method of constructing the “strong” distributionally scrambled set and show that the new dynamical system has a positive topological entropy.
Complexity and proper quasi-weakly almost periodic points
Abstract Zhou and He [1] presented three levels of the orbit’s topological structure. In this paper, we shall try to investigate the orbit’s topological structure of proper quasi-weakly almost periodic point. It is stated that there exists a dynamical system having zero topological entropy such that it has a dense, invariant, extremal and transitive, uncountable distributionally scrambled set consisting of proper quasi-weakly almost periodic points.
The Globalization problem for inner automorphisms and Skolem-Noether theorems
We investigate the circumstances under which an inner automorphism of a ring with local units can be built from “local” information. Specifically, we consider three natural “inner” type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphism is “piecewise” inner. On the other hand, we construct a large class of rings for which every locally inner automorphism is actually inner. Finally we obtain some Skolem-Noether type Theorems for infinite matrix and triangular matrix rings.