Why can we detect the chaos?

A well known chaotic mapping in symbol space is a shift mapping. However, other chaotic mappings in symbol space exist too. The basic change is to consider the process (physical or social phenomenon) not only at a set of times which are equally spaced, say at unit time apart (a shift mapping), but at a set of times which are not equally spaced, say if we cannot fixed unit time (an increasing mapping). Hence we regard x _t is being the flow of discrete signals when t is restricted to values 0,1,2,... but x _f(t) the detection of these signals. Such interpretation simulates the observation. Our results reveal why we can detect chaos even our experiment is not shared in strict equally spaced time intervals. This as every mathematical treatment leads to a rigorous definition of chaos. We restrict ourselves with symbol space A ^w, that is, we consider one sided infinite sequences x ₀, x ₁,..., x _t,... with elements from a fixed set (∀t x _t ∈ A). Our results is proved for such space, namely, the increasing mapping f _ω : A ^ω → A ^ω is chaotic in the set A ^ω, where f _w (x)= x _f(0)x _f(1) x _f(2)... x _f(i)..., i ∈N, x∈A ^ω, 0<f(0) and ∀i∀j[i<j ⇒ f(i)<f(j)].