e−Chaotic Generalized Shift Dynamical Systems

Abstract

In the following text we prove that for bijection ϕ : N → N and discrete set {1, . . . , k} with k ≥ 2, the generalized shift dynamical system ({1, . . . , k} N, σϕ) is e−chaotic, (expansive, positively expansive) if and only if {{ϕ i (n) : i ∈ Z} : n ∈ N} is a finite partition of N (or equivalently there exists N ∈ N such that N =S{ϕi({1, . . . , N}) : i ∈Z}).