Hierarchical Structure of Periodic Orbits of a Hyperbolic Automorphism on the 2-Torus

Abstract

This paper studies the hierarchical structure of periodic orbits of the automorphism induced by the matrix \(A=\begin{pmatrix} 2& 1\\ 1& 1 \end{pmatrix}\) on the torus \({\mathbb {T}}^2\). The induced symbolic dynamics is not trivial with forbidden sequences. We show that the periodic orbits of the system is hierarchically structured by clusters. We establish the number of clusters via symbolic dynamics and digraphs. Algorithms that group all periodic orbits in clusters are given.