 Sharkovskii's theorem

Order the positive integers by odd numbers from $3$ upward, then twice the odds, then four times the odds, and so on, followed by descending powers of two and finally $1$. If a continuous interval map has a cycle of one period, it has cycles of every period later in this order. In particular, period three forces every positive period.

Sharkovskii's theorem is a result in the field of dynamical systems, particularly concerning the behavior of continuous functions on the unit interval \([0, 1]\) and the periodic points of these functions. The theorem provides a remarkable ordering of natural numbers that relates to the existence and types of periodic points in continuous functions.