A Simple Proof of the Chaoticity of Shift Map Under a New Definition of Chaos

Abstract

Recently, Du has given a new strong definition of chaos by using the shift map. In this paper, we give a proof of the main theorem by constructing a dense uncountable invariant subset of the symbol space $\sum_2$ containing transitive points in a simpler way with the help of a different metric. We also provide two examples, which support this new definition.