Last week Boris Kalinin, Victoria Sadovskaya and myself have posted a short preprint on Lyapunov spectrum stability on the arxiv. We have been inspired by work Saghin-Yang discussed here earlier. Namely we have generalized the local rigidity Theorem for Anosov automorphisms on tori by relaxing the assumption on the spectrum. We don’t need the spectrum to be real anymore and allow pairs of complex eigenvalues. We can handle corresponding 2-dimensional intermediate foliations with the help of Continuous Amenable Reduction Theorem (due to Boris and Victoria). Interestingly there is also a partial generalization to certain partially hyperbolic toral automorphisms:
Theorem Let
be a totally irreducible automorphism with exactly two eigenvalues of modulus one and simple real eigenvalues away from the unit circle. Let
be a volume-preserving
-small perturbation of
such that the Lyapunov exponents of
conjugate to
is dense in 
. Based on our computer experiments we conjectured that the strong unstable foliation is transitive and that there is a unique invariant measure with absolutely continuous conditionals on strong unstable leaves (
-Gibbs measure), which is then, of course, the SRB measure. For example the image below shows the points of intersection of the strong unstable manifold of the fixed point and a 
transversal for the Anosov diffeomorphism
be a hyperbolic automophism with real spectrum 
. Let 
be an Anosov diffeomorphism homotopic to A with a partially hyperbolic splitting of the unstable bundle into weak unstable and strong unstable subbundles. Then the strong unstable foliation 
of 
remains unknown as well as the minimality question.
Andrey Gogolev 