Topological dynamics on the interval — Sharkovskii's order

Parent: orbit · Prereq: discrete-dynamical-systems

Setting: $I \subseteq \mathbb{R}$ an interval, $f: I \to I$ continuous (nothing more — no smoothness, no measure). The astonishing fact: mere continuity on a one-dimensional space already forces a rigid arithmetic of periods.

Sharkovskii's ordering

Order the positive integers as follows (⊳ = "precedes"):

Odd numbers first, then $2\times$ odds, then $4\times$ odds, …, and at the very end the powers of two, descending.

Sharkovskii (1964)

If a continuous $f: I \to I$ has a periodic point of period $m$, it has periodic points of every period $n$ with $m \rhd n$.

So period 3 forces all periods (3 is ⊳-maximal), and a map with only finitely many periods can only have periods that are powers of two. The ordering is sharp: for every $m$ there is a map whose set of periods is exactly $\{n : m \rhd n\} \cup \{m\}$.

Li–Yorke (1975), "period three implies chaos"

If $f$ has a period-3 point, there is an uncountable scrambled set $S \subseteq I$: for all $x \neq y$ in $S$,

$$\limsup_{n\to\infty} |f^n(x) - f^n(y)| > 0 \quad\text{and}\quad \liminf_{n\to\infty} |f^n(x) - f^n(y)| = 0.$$

Orbits of scrambled pairs approach each other arbitrarily closely, forever, without ever staying together.

The proof engine behind both: if intervals $J_0, J_1$ satisfy $f(J_0) \supseteq J_0 \cup J_1$ and $f(J_1) \supseteq J_0$ (a covering relation, which a period-3 orbit provides), then every infinite itinerary through the covering graph is realized by some point — an orbit for every symbol sequence. That is symbolic-dynamics entering through the back door.

Count along: a genuine period-3 orbit

Tent map $T(x) = 2\min(x, 1-x)$ on $[0,1]$. Take $x = \tfrac{2}{7}$:

A 3-cycle — so by Sharkovskii the tent map has periodic orbits of every period, and by Li–Yorke it is chaotic.

The one-dimensionality is essential

On the circle, the rotation by $1/3$ has every orbit of period exactly 3 and no other periods at all. Sharkovskii's order is a theorem about the interval's topology (its proof lives on the intermediate value theorem), not about dynamics in general.