Dynamics of shear homeomorphisms of tori and the Bestvina-Handel algorithm
A homeomorphism of a two-dimensional torus is called shear if it is isotopic to one Dehn twist. For a periodic orbit of a shear homeomorphism, the rotation number in \(\mathbb Q\cap [0,1)\) is associated. A special kind of periodic orbits, called simple, is considered. The notion of ``simple pair of two simple periodic orbits'' of a given shear homeomorphism is introduced. Denote by \(r\vee s\) the pair of the rotation numbers \(r\) and \(s\) of two simple periodic orbits which form a simple pair (actually, such \(r\) and \(s\) are Farey neighbors). The following order relation \(\geq\) in the set \(P:=\{r: r\in \mathbb Q\cap [0,1)\}\cup \{r\vee s: r,s\in \mathbb Q\cap [0,1)\}\) is introduced: \(r\vee s\geq t\) if and only if \(t\in [r,s]\) and \(r_1\vee r_2\geq s_1\vee s_2\) if and only if \(s_1,s_2\in [r_1,r_2]\). Resembling slightly the Sharkovskii theorem, the main result of the paper asserts that the existence of a simple pair of periodic orbits with the rotation numbers \(r\vee s\) implies the existence of all simple orbits and simple orbit pairs of rotation numbers smaller than \(r\vee s\) in the order \(\geq\) in \(P\). The proof is based on the Bestvina-Handel algorithm. The result contributes to an explanation of the phenomenom of `quantum accelerator modes' which appear in a physical system called the kicked accelerated particle.
scientific article; zbMATH DE number 6077281
scientific article