Foliations and Conjugacy:Anosov Structures in the plane

TL;DR

This paper explores the classification of Anosov diffeomorphisms on the plane, revealing that preserving stable and unstable manifolds leads to infinitely many conjugacy classes, contrasting with known finite classes.

Contribution

It demonstrates that requiring conjugacies to preserve stable and unstable manifolds results in infinitely many equivalence classes of planar Anosov diffeomorphisms.

Findings

Finite conjugacy classes for linear hyperbolic automorphisms and translations

Infinite classes when stable/unstable manifolds are preserved

Dependence of hyperbolic structure on metric choice

Abstract

In a non-compact setting, the notion of hyperbolicity, and the associated structure of stable and unstable manifolds (for unbounded orbits), is highly dependent on the choice of metric used to define it. We consider the simplest version of this, the analogue for the plane of Anosov diffeomorphisms, studied earlier by W. White and P. Mendes. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations. We show that if the structure of stable and unstable manifolds is required to be preserved by these conjugacies, the number of distinct equivalence classes of Anosov diffeomorphisms in the plane becomes infinite.