Today, we will apply the technical lemma proved yesterday to proving a few special properties of Anosov diffeomorphisms. The first one states that if you have an approximate orbit, then you can approximate it by a real orbit. This may not sound like much, but it is false for isometries—and in fact, it gives another way of proving the structural stability result.

As usual, start with a compact manifold and an Anosov diffeomorphism . We can put a metric on (e.g. by imbedding in euclidean space, or using a Riemannian metric, etc.). To formalize this, fix . We introduce the notion of a -orbit. This is a two sided sequence such that .

Theorem 1 (Anosov shadowing lemma)Fix sufficiently small. There is such that any -orbit can be shadowed by a unique real orbit , i.e. and for all .

To do this, we start with the space and the shift homeomorphism . A double-sided sequence is simply a continuous map of into (since we will give the discrete topology). This sequence is an orbit if and only if we have the equation

Similarly, it is a -orbit if .

But this is familiar. We know from the previous, technical lemma, that if is small, then can be shifted by a small amount to some that satisfies . In particular, this corresponds to a sequence which is a legitimate orbit and such that the are all uniformly small (say ). This completes the proof, since uniqueness follows from the uniqueness in the technical theorem.

In fact, I’m pretty sure we can take to be bounded by a constant multiple of . This is because the construction of the map is Lipschitz continuous with respect to the other parameters (since was constructed out of a rapidly converging iteration process).

Now, let’s do a few examples of why this result is super-useful. First, recall the following definition. A homeomorphism of a compact metric space is called **expansive** if there exists such that for all , we can find some with . This means loosely that no matter how precisely we know , we can never know the entire orbit of to precision less than . (There is a weaker version of this condition, called *sensitive dependence on initial conditions*, as well.)

Theorem 2Anosov diffeomorphisms are expansive.

Indeed, take small, and such that any -orbit is -shadowed by a unique -orbit. Let . Then any two orbits that differ everywhere by distance are two -orbits that -shadow each other; this means that they have to be the same.

Theorem 3Let be an “approximately periodic point” i.e. for small. Then there is an actual periodic orbit, corresponding to a point with , such that is small uniformly in .

This result is left as an exercise to the interested reader. The proof relies on the same technique as in the shadowing lemma, but with the integers modulo replacing the whole integers as the set .

I think I want to blog about something else now, so I’m going to close by saying that, first of all, not that many Anosov diffeomorphisms are known! The basic example (which I really, really ought to have mentioned earlier) is given by a hyperbolic automorphism (with integer entries in the matrix and determinant ) of the torus. Also, Anosov diffeomorphisms are just the first part of a long story; basically, they are “uniformly hyperbolic” (in that the constants for dilation and contraction are constant). There is a whole theory of nonuniform hyperbolicity, called Pesin theory—and y’know, Yakov Pesin is actually teaching our next course, on fractal geometry and dynamics–which I don’t know about.

July 15, 2010 at 5:51 pm

[…] example of an expansive homeomorphism is given by any Anosov diffeoorphism, as we discussed—this is a corollary of the uniqueness in the shadowing lemma. There are actually no expansive […]

July 18, 2010 at 1:08 pm

[…] a “near orbit” can be approximated very closely by a legitimate periodic orbit by the Anosov closing lemma, and we know that the sum of over a legitimate periodic orbit is exactly zero. This is the whole […]