At the third conference on Dynamics of Differential Equations, I explained the following Spectral Decomposition Theorem for arbitrary smooth surface diffeomorphisms. It is mostly part of the joint work with Sylvain Crovisier and Omri Sarig.
Given some dynamics, say that if the symmetric difference of the two subsets does not carry measures with positive entropy.
Theorem (B-Crovisier-Sarig). For a
diffeomorphism of a closed surface, let
be its infinite homoclinic classes. Then the non-wandering set satisfies:
with
and
topologically transitive. More precisely, for each
, there is a compact set
and an integer
such that
is topologically mixing,
,
and
if
.
If
is topologically transitive and
, then there is a unique infinite homoclinic class. If, additionally,
is topologically mixing, then this homoclinic class is itself topologically mixing.
I also explained that we have a good understanding of the dynamics inside the pieces (measures of maximum entropy, periodic orbits, Borel classification and the periods involved in the three descriptions are the above period ). See these slides for details.