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…)