Classification of Anosov diffeomorphisms

Classify Anosov diffeomorphisms up to topological conjugacy, extending the known classifications for codimension-one systems and systems on nilmanifolds.

Background

The paper places its results in the broader context of Smale’s 1967 conjectural classification of completely hyperbolic systems. Expanding local diffeomorphisms have been classified up to topological conjugacy as affine expanding endomorphisms of infra-nilmanifolds, but the corresponding classification problem for Anosov diffeomorphisms remains unresolved.

The paper notes two important partial results: codimension-one Anosov diffeomorphisms force the underlying manifold to be a torus, while Anosov diffeomorphisms on nilmanifolds are topologically conjugate to affine hyperbolic automorphisms. A general classification beyond these settings is therefore explicitly identified as open.