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 {M} and an Anosov diffeomorphism {f: M \rightarrow M}. We can put a metric {d} on {M} (e.g. by imbedding {M} in euclidean space, or using a Riemannian metric, etc.). To formalize this, fix {\delta>0}. We introduce the notion of a {\delta}-orbit. This is a two sided sequence {\{x_n\}_{n \in \mathbb{Z}}} such that {d(x_{n+1},  f(x_n)) \leq \delta}.

Theorem 1 (Anosov shadowing lemma) Fix {\epsilon>0} sufficiently small. There is {\delta>0} such that any {\delta}-orbit {\{x_n\}} can be shadowed by a unique real orbit {\{y_n\}}, i.e. {y_{n+1} = f(y_n)} and {d(x_n, y_n) <  \epsilon} for all {n \in  \mathbb{Z}}.

To do this, we start with the space {X=\mathbb{Z}} and the shift homeomorphism {h: n \rightarrow  n+1}. A double-sided sequence is simply a continuous map {\phi} of {\mathbb{Z}} into {M} (since we will give {\mathbb{Z}} the discrete topology). This sequence is an orbit if and only if we have the equation

\displaystyle  f \circ \phi = \phi \circ h.

Similarly, it is a {\delta}-orbit if {d(f  \circ \phi, \phi \circ h)<\delta}.

But this is familiar. We know from the previous, technical lemma, that if {\delta} is small, then {\phi} can be shifted by a small amount to some {\psi: \mathbb{Z} \rightarrow M} that satisfies {f  \circ \psi = \psi \circ h}. In particular, this corresponds to a sequence {\{y_n\}} which is a legitimate orbit and such that the {d(y_i,x_i)} are all uniformly small (say {<\epsilon}). This completes the proof, since uniqueness follows from the uniqueness in the technical theorem.

In fact, I’m pretty sure we can take {\delta} to be bounded by a constant multiple of {\epsilon}. This is because the construction of the map {\psi} is Lipschitz continuous with respect to the other parameters (since {\psi} 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 {f: X  \rightarrow X} of a compact metric space is called expansive if there exists {\eta>0} such that for all {x \neq y}, we can find some {n \in  \mathbb{Z}} with {d(f^n(x), f^n(y)) \geq  \eta}. This means loosely that no matter how precisely we know {x}, we can never know the entire orbit of {x} to precision less than {\eta}. (There is a weaker version of this condition, called sensitive dependence on initial conditions, as well.)

Theorem 2 Anosov diffeomorphisms are expansive.

Indeed, take {\epsilon>0} small, and {\delta>0} such that any {\delta}-orbit is {\epsilon}-shadowed by a unique {\epsilon}-orbit. Let {\eta = \min(\delta,  \epsilon)}. Then any two orbits that differ everywhere by distance {<\eta} are two {\delta}-orbits that {\epsilon}-shadow each other; this means that they have to be the same.

Theorem 3 Let {x} be an “approximately periodic point” i.e. {d(T^nx,  x)<\delta} for {\delta} small. Then there is an actual periodic orbit, corresponding to a point {p \in X} with {T^n p = p}, such that {d(T^ix, T^ip)} is small uniformly in {i}.

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 {n} replacing the whole integers as the set {X}.

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 \pm 1) 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.