A major part of dynamical system theory studies the iteration of diffeomorphisms under various assumptions, geometric or analytic or otherwise. It is also interesting to study them collectively. In particular, , the set of diffeomorphisms of some manifold and some number , is a group. It is for instance of interest to know when it is simple.
Todd Fisher gave a talk in Orsay about the (non)triviality of the centralizer for many elements. Recall that the centralizer of an element of a group is , the set of all elements that commute with . It is also the set of all elements that conjugate with itself, i.e., it can be interpreted as the set of symmetries of .
may be very large, e.g., equal to the whole of if this group is Abelian. It is said to be trivial if it is reduced to (which is always part of it). Note that such a diffeomorphism cannot be embedded in a smooth flow.
One expects that the set of -diffeomorphisms of with trivial centralizer is very big (residual or even open and dense). To show this for all compact manifolds and all ) was one of Smale’s questions for the the twenty-first century.
Results are known in dimension one or -smoothness under hyperbolicity assumptions:
- is open and dense for all (Kopell);
- a generic diffeomorphism of an arbitrary manifold has trivial -centralizer (Bonatti, Crovisier, Wilkinson 2008).
It has turned out that the above is really only generic and fails to hold on an open and dense set according to Bonatti, Crovisier, Vago and Wilkinson. On the circle, this holds only on a set with empty interior. This last result extends to tori in dimensions up to 4 according to a work in progress of Bakker and Fisher.
Hyperbolicity also allows some results:
- a generic -axiom A diffeomorphisms with no cycles has trivial centralizer (Tugawa 1978);
- there is an open and dense set of surface axiom-A diffeomorphisms with no cycles having a trivial -centralizer (Palis-Yoccoz 1989). This even holds for arbitrary dimensions if one assumes the existence of a sink or a source.
On surfaces, Fisher has extended the last Palis-Yoccoz result to intermediate smoothness, i.e., , and axiom A with no-cycles, i.e., -stable, diffeomorphisms instead of the above ones with strong transversality, i.e., the structurally stable ones.
The proof of Palis and Yoccoz (and of its later generalizations) relies on a local theorem showing that on each basin, two commuting maps must be iterates one of the other and a global theorem that connects the basins and identify the iterates. The last is the delicate part of the proof.