  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.
