A refinement of Šarkovskiĭ’s theorem

Abstract:

Let $f:R \to R$ be continuous. If $f$ has an orbit of period $n$, the question of which other periods $f$ must necessarily have was answered by A. N. Sarkovskii by giving a total ordering of the natural numbers, now called the Sarkovskii ordering. The ordering does not take into account the period types and examples show that depending on the type of the period other periods than those implied by the Sarkovskii ordering are present. Introducing the concepts of a periodic loop (a periodic orbit of a certain type) and infinite loop, we give a total ordering of loops and obtain, as a consequence, a refinement of the theorem of Sarkovskii.