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 
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,
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 
est un système dynamique probabiliste constitué de l’espace 
muni de sa tribu borélienne et d’une mesure produit 
et du décalage 
. Un isomorphisme entre systèmes dynamiques probabilistes 
est une bijection 
vérifiant 
et 
.
sont isomorphes si et seulement s’ils ont la même entropie: 
(
sont les vecteurs de probabilité définissant les deux décalages; 
par convention).
). L’ensemble 
des couplages ergodiques est un 
, donc encore un espace de Baire. L’énoncé principal de Downarowicz et Serafin est:
alors l’ensemble des facteurs 
est une partie générique de 
, l’ensemble des 
-facteurs 
tels que pour tout 
, il existe un borélien 
tel que la différence symétrique entre 
et 
soit de 
étant manifestement ouverts et emboîtés, il suffit de montrer que tout couplage ergodique 
pour 
(
)
and 
for some 
, the finite sequences being considered up to translation of the indices. The self-map 
(called the shift) turns 
into a dynamical system.
.
, some of the most natural questions are:
where 
is the set of all the finite sequences of length $n$ which occurs as a subsequence of any 
and 
is the cardinality of that set.
, which is not included in a larger one.
-matrix with 0,1-entries. The largest positive eigenvalue is the exponential of the entropy (or zero if and only if 
) and its multiplicity is the number of irreducible components. But of course 
is very large if 
and a subshift of finite type 
is defined by declaring that each sequence of length belongs to 
. He then considers the limit of the probabilities when 
. Let us quote his main results in a somewhat weakened form and specialized to 
for simplicity.
(empty shift with positive limit probability, zero entropy with full limit probability, non-trivial distribution of the number of components) and large 
and a single, aperiodic irreducible component with full limit probability).
. For large 
, the convergence of the entropy for 
and the irreducibility and aperiodicity for 
where 
is a continuous self-map and 
where 
is a closed subset of 
and 
such that 
.
and a continuous surjection 
commuting with the dynamics: 
. There is an obvious one: a symbolic system (and therefore its topological factors) has finite topological entropy. Is there any other?
map have no symbolic extension whatever, leaving open the question of diffeomorphisms with finite smoothness.
interval maps also always have symbolic extensions for 
.
surface diffeomorphisms, see his preprint 
on 
, small numbers 
such that, setting 
, for all 
:
;
.
is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. 
on the other hand is shown to admit 2 disjoint closed quadrilaterals 
such that 
hyperbolically crosses both 
and 
and 
hyperbolically crosses 
be a C^2 diffeomorphism of a compact manifold which preserves volume. Let 
be a hyperbolic periodic point. Define 
has a transverse point 
. Define 
similarly.
and 
then these two sets are equal (up to a zero volume set) and the restriction of 
for 
. If the exponents along the fiber vanish, then these holonomy preserve the conditional measures. If the base map is hyperbolic and the base measure has a product structure then the conditional measures depend continuously on the base point.
be partially hyperbolic with a central foliation into circles. If 