A Profinite Group Invariant for Hyperbolic Toral Automorphisms

TL;DR

This paper introduces a new profinite group invariant derived from homoclinic groups and periodic data to classify hyperbolic toral automorphisms up to conjugacy.

Contribution

It constructs a profinite group invariant with a module structure that fully characterizes conjugacy classes of hyperbolic toral automorphisms.

Findings

Profinite group invariant distinguishes conjugate automorphisms.

Homoclinic groups and periodic data are key to classification.

Provides a complete topological classification method.

Abstract

For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic left action periodic data, that the homoclinic groups of their right actions play the key role in determining whether or not they are conjugate. This gives a complete set of dynamically significant invariants for the topological classification of hyperbolic toral automorphisms.