📂Dynamics

Sharkovsky's Theorem

Theorem ¹

$$3 \prec 5 \prec 7 \prec 9 \prec \cdots \prec \\ 2\cdot 3 \prec 2 \cdot 5 \prec \cdots \prec \\ 2^2 3 \prec 2^2 5 \prec \cdots \prec \\ 2^3 3 \prec 2^3 5^2 \prec \cdots \prec \\ 2^3 \prec 2^2 \prec 2^1 \prec 2^0$$ The order mentioned above for a transitive relation $\prec$ is known as Sharkovskii’s ordering. Let’s say a continuous map $f : \mathbb{R} \to \mathbb{R}$ has a periodic-$p$ orbit. If $p \prec q$, then $f$ will have a periodic-$q$ orbit.

Explanation

Sharkovskii’s Theorem generalizes the Li-Yorke Theorem, stating that if there exists a periodic-$3$ orbit, not only does a periodic-$m$ orbit exists for all natural numbers $m$, but it also guarantees the existence of a periodic-$q$ orbit for any $p$ following Sharkovskii’s ordering. The ‘beginning’ of Sharkovskii’s ordering is $3$, so if a periodic-$3$ orbit exists, it ensures the existence of all periodic orbits, fully covering the Li-Yorke theorem.

However, in reality, Li-Yorke’s paper was published in 1975, and Sharkovskii’s paper in 1964, so it is more accurate to consider the Li-Yorke theorem as a corollary of Sharkovskii’s theorem. Due to the Cold War era, Sharkovskii’s findings were known to the world much later, and by the time it was disclosed, the Li-Yorke theorem had already become a central theorem in chaos theory.