Periodic orbits of linear endomorphisms on the 2-torus and its lattices

TL;DR

This paper investigates the counting of periodic orbits for linear endomorphisms on the 2-torus, exploring the connection between global and local dynamics and relating the dynamical zeta functions on the torus and finite lattices.

Contribution

It provides a complete classification of lattice endomorphisms up to local conjugacy using invariants like determinant, trace, and a third matrix invariant.

Findings

Explicit relation between zeta functions on the torus and lattices

Complete determination of lattice endomorphisms by key invariants

Insights into the interplay between global and local dynamical properties

Abstract

Counting periodic orbits of endomorphisms on the 2-torus is considered, with special focus on the relation between global and local aspects and between the dynamical zeta function on the torus and its analogue on finite lattices. The situation on the lattices, up to local conjugacy, is completely determined by the determinant, the trace and a third invariant of the matrix defining the toral endomorphism.