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.