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 ${n \in \mathbb{Z}}$. (more…)