Implication diagram of five chaos characterizations: a survey on compact metric space and shift of finite type

Abstract

We give a summary on the development of the relationships between some chaos characterizations, focusing on Devaney chaos, locally eventually onto, strong dense periodic points, topologically mixing and totally transitive. We represent the relationships in four implication diagrams. We look at compact metric spaces in general and on shifts of finite type in particular. At the end, it is shown that different spaces yield different results.