A survey on ergodicity of Anosov diffeomorphisms

This is in part a preparation for my 25-minutes talk in a workshop here at Princeton next week. (Never given a short talk before…I’m super nervous about this >.<) In this little survey post I wish to list some background and historical results which might appear in the talk.

Let me post the (tentative) abstract first:


Title: Volume preserving extensions and ergodicity of Anosov diffeomorphisms

Abstract: Given a C^1 self-diffeomorphism of a compact subset in \mathbb{R}^n, from Whitney’s extension theorem we know exactly when does it C^1 extend to \mathbb{R}^n. How about volume preserving extensions?

It is a classical result that any volume preserving Anosov di ffeomorphism of regularity C^{1+\varepsilon} is ergodic. The question is open for C^1. In 1975 Rufus Bowen constructed an (non-volume-preserving) Anosov map on the 2-torus with an invariant positive measured Cantor set. Various attempts have been made to make the construction volume preserving.

By studying the above extension problem we conclude, in particular the Bowen-type mapping on positive measured Cantor sets can never be volume preservingly extended to the torus. This is joint work with Charles Pugh and Amie Wilkinson.


A diffeomorphism f: M \rightarrow M is said to be Anosov if there is a splitting of the tangent space TM = E^u \oplus E^s that’s invariant under Df, vectors in E^u are uniformly expanding and vectors in E^s are uniformly contracting.

In his thesis, Anosov gave an argument that proves:

Theorem: (Anosov ’67) Any volume preserving Anosov diffeomorphism on compact manifolds with regularity C^2 or higher on is ergodic.

This result is later generalized to Anosov diffeo with regularity C^{1+\varepsilon}. i.e. C^1 with an \varepsilon-holder condition on the derivative.

It is a curious open question whether this is true for maps that’s strictly C^1.

The methods for proving ergodicity for maps with higher regularity, which relies on the stable and unstable foliation being absolutely continuous, certainly does not carry through to the C^1 case:

In 1975, Rufus Bowen gave the first example of an Anosov map that’s only C^1, with non-absolutely continuous stable and unstable foliations. In fact his example is a modification of the classical Smale’s horseshoe on the two-torus, non-volume-preserving but has an invariant Cantor set of positive Lebesgue measure.

A simple observation is that the Bowen map is in fact volume preserving on the Cantor set. Ever since then, it’s been of interest to extend Bowen’s example to the complement of the Cantor set in order to obtain an volume preserving Anosov diffeo that’s not ergodic.

In 1980, Robinson and Young extended the Bowen example to a C^1 Anosov diffeomorphism that preserves a measure that’s absolutely continuous with respect to the Lebesgue measure.

In a recent paper, Artur Avila showed:

Theorem: (Avila ’10) C^\infty volume preserving diffeomorphisms are C^1 dense in C^1 volume preserving diffeomorphisms.

Together with other fact about Anosov diffeomorphisms, this implies the generic C^1 volume preserving diffeomorphism is ergodic. Making the question of whether such example exists even more curious.

In light of this problem, we study the much more elementary question:

Question: Given a compact set K \subseteq \mathbb{R}^2 and a self-map f: K \rightarrow K, when can the map f be extended to an area-preserving C^1 diffeomorphism F: \mathbb{R}^2 \rightarrow \mathbb{R}^2?

Of course, a necessary condition for such extension to exist is that f extends to a C^1 diffeomorphism F (perhaps not volume preserving) and that DF has determent 1 on K. Whitney’s extension theorem gives a necessary and sufficient criteria for this.

Hence the unknown part of our question is just:

Question: Given K \subseteq \mathbb{R}^2, F \in \mbox{Diff}^1(\mathbb{R}^2) s.t. \det(DF_p) = 1 for all p \in K. When is there a G \in \mbox{Diff}^1_\omega(\mathbb{R}^2) with G|_K = F|_K?

There are trivial restrictions on K i.e. if K separates \mathbb{R}^2 and F switches complementary components with different volume, then F|_K can never have volume preserving extension.

A positive result along the line would be the following slight modification of Moser’s theorem:

Theorem: Any C^{r+1} diffeomorphism on S^1 can be extended to a C^r area-preserving diffeomorphism on the unit disc D.

For more details see this pervious post.

Applying methods of generating functions and Whitney’s extension theorem, as in this paper, in fact we can get rid of the loss of one derivative. i.e.

Theorem: (Bonatti, Crovisier, Wilkinson ’08) Any C^1 diffeo on the circle can be extended to a volume-preserving C^1 diffeo on the disc.

With the above theorem, shall we expect the condition of switching complementary components of same volume to be also sufficient?

No. As seen in the pervious post, restricting to the case that F only permute complementary components with the same volume is not enough. In the example, K does not separate the plane, f: K \rightarrow K can be C^1 extended, the extension preserves volume on K, and yet it’s impossible to find an extension preserving the volume on the complement of K.

The problem here is that there are ‘almost enclosed regions’ with different volume that are being switched. One might hope this is true at least for Cantor sets (such as in the Bowen case), however this is still not the case.

Theorem: For any positively measured product Cantor set C = C_1 \times C_2, the Horseshoe map h: C \rightarrow C does not extend to a Holder continuous map preserving area on the torus.

Hence in particular we get that no volume preserving extension of the Bowen map can be possible. (not even Holder continuous)

A few interesting items from the ICM

So, as many people know, as part of my India vacation, I went to the ICM in Hyderabad.

On this last day of conference, I decided to write a small note of a few cool items I picked up in some talks: (although there are in fact many, many other cool facts I might write more once I got back :-P)

1. Renormalization (on Artur Avila‘s talk) So we look at one-dimensional systems, one may zoom in at a part of the interval that maps into itself, which yields a similar (or not) system, the idea is called ‘renormalization’. i.e. we have the ‘renormalization operator’ acting on a certain class of systems (‘renormalizable systems’) and this gives a map on the function space. Now we study the dynamics there(!) At the first glance, it doesn’t look like solving a one-dimensional problem in infinite dimensional space would help in any useful way, but it does(!). As an example, we look at space of circle diffeomorphisms, they have a rotation number, it’s not hard to see, in this case, the linear rotations form a circular attractor for the renormalization operator, further more, the dynamics is perfectly understood on the circle (Gauss map), the operator permutes (infinite dimensional) fibers with equal rotational number. It turns out we know enough about the dynamics on the function space to get useful information to the original problem! Super cool~

2. Differentiating Lipschitz functions and decomposing Kakeya sets (on Marianna Csornyei‘s talk, for details please refer to their ICM paper) They had a through study of exactly which sets in \mathbb{R}^n can be contained in the discontinuity set of a Lipschitz function f: \mathbb{R}^n \rightarrow \mathbb{R}^m. I found the following unbelievable at the first glance: Given a cone C in \mathbb{R}^n (a set of rays from \bar{0}), the C-width of set E \subseteq \mathbb{R}^n is, roughly speaking, the \sup of lengths of E \cap \gamma where \gamma is a Lipschitz curve going only in directions in C. (A more precise definition requires a generalized notion of ‘tangent’ for Lipschitz curves and can be found in the paper). They proved that:

Theorem: Any Lebesgue 0 set in \mathbb{R}^2 can be decomposed into two sets A and B that A has C-width 0 for C= [0, \pi/2] and B has C'-width 0 for C' = [\pi/2, \pi].

Why does this surprise me? Well, of course the first thing I donsider is: what would happen for the Kakeya set? Our null set contains a line segment in each direction, hence even if we just requiring the decomposed sets A and B to intersect all straight lines in directions of C, C' in length 0 sets would give pretty much only one possible decomposition: we have to take A to be the union of all segments in direction of C', each missing a linear 0 set, and same for B and C(! not much freedom, right?). It’s already hard to believe such A and B can be made satisfying the property, not to mention that in fact they can be made intersecting all Lipschitz curves in null sets (!) (At first I thought it was an obvious counterexample to the theorem, but after discussing with her after the talk, this is indeed what the theorem does) Amazing…

List to be filled in:

3. Boundary rigidity via filling volume (On Sergei Ivanov‘s talk. For details please refer to his paper on the ArXiv)

4. Constant main curvature surfaces

…to be continued…