An Anosov diffeomorphism is a 
diffeomorphism 
of a manifold 
to itself such that the tangent
bundle of 
is hyperbolic with respect to 
. Very few classes of Anosov diffeomorphisms are known. The
best known is Arnold's cat map.
A hyperbolic linear map 
with integer entries
in the transformation matrix and determinant 
is an Anosov diffeomorphism of the
-torus. Not every manifold admits
an Anosov diffeomorphism. Anosov diffeomorphisms are expansive,
and there are no Anosov diffeomorphisms on the circle.
It is conjectured that if 
is an Anosov diffeomorphism on a compact Riemannian manifold and the nonwandering set 
of 
is 
, then 
is topologically
conjugate to a finite-to-one factor of
an Anosov automorphism of a nilmanifold.
It has been proved that any Anosov diffeomorphism on the 
-torus is topologically
conjugate to an Anosov automorphism, and
also that Anosov diffeomorphisms are 
structurally stable.
