Unimodal orbits in continuous functions and generalization of Sarkovskii's theorem (Q1176054)

In 1964 Sharkovskij obtained a theorem about the relations between the periods for the existence of the periodical orbits of continuous functions on a compact interval. Let \(f\) be a function on \(X\subset\mathbb{R}\). If there exists \(\tau\in X\) such that \(f(x)\leq f(y)\) for all \(x\) and \(y\) with \(x<y<\tau\) or \(\tau<y<x\), then \(f\) is called a unimodal function on \(X\) relative to \(\tau\). The paper finds some relations between orbits of the unimodal functions. These orbits are periodic and nonperiodic, finite and infinite. Moreover, a classification of the space of continuous functions on a compact interval is given.