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.
Zhang (2026) reported that every holomorphic Anosov diffeomorphism of complex codimension one on a compact, connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. Here complex codimension one means that either the stable or the unstable subbundle has complex dimension one. The statement covers the complex-codimension-one case of Ghys's conjecture (Ghys 1995), including holomorphic Anosov diffeomorphisms in complex dimension 3. The general holomorphic conjecture remains open.
Zhang (2026) credits Intern-S2 with assistance in developing the mathematical argument and reports a Lean formalization produced with SHEAF. The formalized classification assumes an established theorem on the topological rigidity of homotopy tori, rather than formalizing that theorem. Independent specialist review of the new argument had not been reported as of Oct. 15, 2026.
