Period 3 implies chaos

10 Feb 2019

Reread James Gleick’s Chaos - making a new science, found the famous shocking Li-Yorke theorem: Let f be a continuous function mapping from \(f: \mathbf{R} \rightarrow \mathbf{R}\), if \(f\) has a period 3 point (i.e. \(f^3(x) = x\) and \(f(x), f^2(x) \neq x\)), then

1. For every \(k = 1,2,...\) there is a periodic point having period \(k\).

2. There is an uncountable set S containing no period points, which satisfies

• For every \(p,q \in S\), \(p \neq q\),

\[\lim_{n\rightarrow \infty} \sup |F^n(p) - F^n(q)| > 0\] \[\lim_{n\rightarrow \infty} \inf |F^n(p) - F^n(q)| = 0\]

• For every \(p \in S\) and a periodic point \(q \in \mathbf{R}\),

\[\lim_{n\rightarrow \infty} \sup |F^n(p) - F^n(q)| > 0\]

This depicts a bizarre behavior from a simple continuous function! Once a period 3 point is found, then there are points with any arbitrary periods; and uncountable many chaotic points, which can approach to each other then departure, and can approach to any periodic points as if it’s converging but suddenly jump closer to another periodic point.