Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

TL;DR

This paper explores the dynamics, orbit structures, and symmetries of integer matrix-induced transformations on rational lattices of tori, with implications for quantum chaos and automorphism symmetries.

Contribution

It provides a detailed analysis of orbit structures and symmetries of non-invertible and invertible toral endomorphisms on rational lattices, extending understanding of their dynamical properties.

Findings

Characterized pretails of eventually periodic orbits.

Analyzed symmetries and reversing symmetries of toral automorphisms.

Relevance to quantum cat maps and automorphism symmetry classification.

Abstract

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.