Chaos and One-to-Oneness

Abstract

The term was introduced in a mathematical context by T. Li and J. Yorke [6]. This paper considers chaos in the setting of an interval I of real numbers and a function f that maps I into I. Such a pair (I, f) is called a dynamical system, or simply a system. Given a system (I, f), the function x -*f(f(x)) also maps I into I and is denoted by f[2]. More generally, the function x >f(f(. . . f(f( x) ... ))), where f appears n times, maps I into I, is denoted by f[n], and is called the nt/ iterate off, f[l] being f itself. It is sometimes useful to think of I as the set of states of a physical system. Then f(x) may be thought of as the state of the system that results from an initial state x after one unit of time. Thus, f [n](x) is the state of the system after n time units, given that the initial state was x. A sequence of states x,f(x), f[2](X),...,f[ln](X),... describes the evolution of the system over time. Whether the system (I,f) behaves chaotically is related to the behavior of the sequence of iterates f, f[2],f[3],... f[n] Roughly speaking, in the system means that the iterates of f churn up the points of I. For chaos to occur, this churning up process should not occur separately in disjoint parts of I. A function f is one-to-one if f(x) =f( y) only if x = y. We will prove that, for a one-to-one function f on I, chaotic behavior cannot occur if f is continuous, but that a type of chaotic behavior may occur if f has even one discontinuity. Although it is not the main concern of this paper, we note in passing that continuous functions that are not one-to-one-even seemingly harmless ones-may also lead to chaos.