A geometric algorithm to decide the forcing relation on cycles (Q1978902)

Let \(f:\mathbb R\rightarrow \mathbb R\) be a continuous map, and \(x_1 < \ldots < x_s\) elements of a periodic orbit orb\(_f(x_1)\) with least period \(s\). In more detail, if \(\eta : \{1,\ldots ,s\}\rightarrow \{1,\ldots ,s\}\) is a bijection satisfying \(\eta ^k(1)\not = 1\) for \(1\leq k < s\) such that \(f(x_i)=x_{\eta (i)}\) then orb\(_f(x_1)\) is said to have the orbit type \(\eta \) (\(x_1\) can be replaced by any \(x_i\) here). Then the forcing relation is defined as follows: \(\vartheta \) forces \(\eta \) if every continuous map \(f\) that has an orbit of type \(\vartheta \) also has an orbit of type \(\eta \). The author presents a new and rather efficient algorithm to determine whether one cycle forces another. It is based on the machinery of oriented graphs and in fact provides much more, namely a certain clarification of the structure of the partial order relation induced by the forcing.