Dynamical systems and ergodic theory Topological dynamics Theorem

Period three implies chaos

Builds on Interval Continuous map Periodic point Sharkovskii's theorem

Statement

Let $J \subseteq \R$ be an interval and let $F \colon J \to J$ be a continuous map. Suppose there is a point $a \in J$ such that $b = F(a)$ , $c = F^2(a)$ and $d = F^3(a)$ satisfy $d \le a < b < c$ or $d \ge a > b > c$ . Then:

For every integer $k \ge 1$ there is a periodic point in $J$ of minimal period $k$ .
There is an uncountable set $S \subseteq J$ that contains no periodic point and satisfies:
- for all $p, q \in S$ with $p \neq q$ , $\limsup_{n \to \infty} \abs{F^n(p) - F^n(q)} > 0$ and $\liminf_{n \to \infty} \abs{F^n(p) - F^n(q)} = 0$ ;
- for every $p \in S$ and every periodic point $q \in J$ , $\limsup_{n \to \infty} \abs{F^n(p) - F^n(q)} > 0$ .

Remarks

This is the theorem of Li and Yorke, from their paper "Period three implies chaos". A point of minimal period $3$ satisfies the hypothesis after a suitable choice of $a$ on its orbit: one of the three points can serve as $a$ , and then $d = a$ . The proof of part 1 uses a covering form of the intermediate value theorem. If $F(K) \supseteq L$ for compact intervals $K, L$ , then $K$ contains a compact interval mapped onto $L$ , and chains of such intervals give periodic points. Part 1 is also a special case of Sharkovskii's theorem, since $3$ is first in the Sharkovskii ordering. Example: the logistic map $F(x) = 4x(1 - x)$ on $[0, 1]$ has the period- $3$ orbit of $\sin^2(\pi/7)$ , $\sin^2(2\pi/7)$ , $\sin^2(4\pi/7)$ , so it has points of all periods. The interval hypothesis is necessary: the rotation of a circle by one third has period $3$ points but no fixed point.

Going deeper

For a student

The result gives an easy test for complicated dynamics in one-dimensional maps: find one point with the stated order pattern. The set $S$ is called a scrambled set, and maps with an uncountable scrambled set are called chaotic in the sense of Li and Yorke. The theorem says nothing about how large $S$ is in measure; for some maps with period three, almost every orbit tends to a stable periodic orbit.

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Tag	`93VXAB8JYS`
URI	`m2://93VXAB8JYS`
Kind	statement, theorem
Writing	W0 stub
MSC	37E05, 37D45
Status	This copy of the site has no build data. Run `m2 status 93VXAB8JYS`.
About	Periodic point
Uses	Interval, Continuous map, Periodic point, Sharkovskii's theorem