Andrey Gogolev, "Moduli of smooth conjugacy
of Anosov diffeomorphisms of Tn, n>2":
Structural stability asserts that if two Anosov diffeomorphisms are close
enough then they are conjugate: hf=gh. It's known that the conjugacy h is
Hölder continuous.
There are simple obstructions for h to be smooth. Let p be a periodic point
of f, fn(p)=p, g^n(h(p))=h(p). If h were differentiable, then
the differentials D(fn)(p) and D(gn)(h(p)) would be conjugate by
the differential of h, and we say that f and g have same periodic data.
De la Llave showed that if f and g are Anosov diffeomorphisms of T2, then
conjugacy is smooth if f and g have the same periodic data. For
diffeomorphisms of Tn, n>3, the corresponding statement doesn't hold --
periodic data is not a complete set of invariants for smooth conjugacy.
We consider Anosov diffeomorphisms of Tn, n>2. Under some additional
assumptions we prove that the conjugacy h is smooth if periodic data of f and
g coincide.
In case n=3 we require that the Mather spectra of f and g have 3 connected
components -- f and g can be viewed as partially hyperbolic
diffeomorphisms.
This is joint work with Misha Guysinsky.