Anosov closing lemma | Climbing Mount Bourbaki

We continue the discussion of the cohomological equation started yesterday and explain a situation where it can be solved.

Given a compact space ${X}$ , a continuous map ${T: X \rightarrow X}$ , and a continuous ${g: X \rightarrow \mathbb{R}}$ with vanishing periodic data as in yesterday’s post, there are not many ways to construct a solution ${f}$ of the cohomological equation

$\displaystyle g = f \circ T - f.$

The basic thing to note that if ${f(x)}$ is known, then recursively we can determine ${f}$ on the entire orbit of ${x}$ in terms of ${g}$ . In case the map ${T}$ is topologically transitive, say with a dense orbit generated by ${x_0}$ , then by continuity the entire map ${f}$ is determined by its value on ${x_0}$ .

This also provides the method for obtaining ${f}$ in the topologically transitive case. Namely, one picks ${f(x_0)}$ aribtrarily, defines ${f(T^ix_0)}$ in the only way possible by the cohomological equation. In this way one has ${f}$ defined on the entire orbit ${T^{\mathbb{Z}}(x_0)}$ such that on this orbit, the equation is satisfied. If one can show that ${f}$ is uinformly continuous on ${T^{\mathbb{Z}}(x_0)}$ , then it extends to the whole space and must by continuity satisfy the cohomological equation there too.

This is the strategy behind the proof of the theorem of Livsic from the seventies, whose proof we shall sketch:

Theorem 1 (Livsic) Let ${M}$ be a compact Riemannian manifold, ${T: M \rightarrow M}$ a topologically transitive Anosov diffeomorphism. If ${g: M \rightarrow \mathbb{R}}$ is an ${\alpha}$ -Holder function such that ${T^n p =p}$ implies ${\sum_{i=0}^{n-1} g(T^i p) = 0}$ , then there exists an ${\alpha}$ -Holder ${f: M \rightarrow \mathbb{R}}$ such that

$\displaystyle g = f \circ T -f .$

(more…)

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…)

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Climbing Mount Bourbaki

The Livsic theorem on the cohomological equation

A few more facts about Anosov diffeomorphisms

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