Periodic Points of the Open-Tent Function
Danrun Huang, Daniel J. Scully · Mathematics Magazine · 2003
Given a function f : S + S, it is of great interest in the field of dynamical systems to figure out which points in the set S are eventually sent back to themselves through repeated applications of f . More precisely, people like to know which points x and which positive integers n have the property that f n (x) = x , where f n denotes the nth iteration o f f . Such a point x is called aperiodicpoint, and the smallest such n is called the prime period of x. (Note that this does not require that n be a prime number.) According to a famous theorem by Li and Yorke [ti],for a continuous function f on a line or a closed interval S . if f has a point of prime period 3, then f has a point of prime period n for every n. This amazing result turns out to be a special case of the even more amazing Sarkovskii theorern [4. Ch. 111. To construct a simple example of a continuous function with a point of prime period 3 on the unit interval, we choose 0 -t 112 -t l -+ 0 as our 3-cycle and connect the points (0, 112). (1 12, 1). and ( 1 ,0) by a piecewise-linear function