The Livsic theorem on the cohomological equation

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

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

As already discussed, the proof begins by choosing ${x_0 \in X}$ such that the orbit is dense, and defining

$\displaystyle f(T^n x_0) =\sum_{i=0}^{n-1} g(T^i _0), \quad n \in \mathbb{Z}.$

The point is now to prove that if ${T^n x_0}$ and ${T^m x_0}$ are close, then ${f(T^n x_0), f(T^m x_0)}$ are close. Now, the closeness of ${T^n x_0, T^m x_0}$ is the same as saying that ${T^n x_0}$ is “almost periodic” with period ${m-n}$ (say ${n<m}$ for definiteness). The difference ${f(T^n x_0) -f(T^m x_0)}$ is the sum of ${g}$ over the near-orbit

$\displaystyle \{ T^n x_0, T^{n+1} x_0, \dots, T^{n+m - 1} x_0 \}.$

So, if we show that the sum of ${g}$ over a “near orbit” is small, then we will have shown the continuity of ${f}$ . The justification is that a “near orbit” can be approximated very closely by a legitimate periodic orbit by the Anosov closing lemma, and we know that the sum of ${g}$ over a legitimate periodic orbit is exactly zero. This is the whole point of the proof of the Livsic theorem.

We shall prove that there exist ${C, \delta>0}$ such that if ${d(T^n x, x) < \delta}$ for any ${x \in X, n \in \mathbb{N}}$ , then

$\displaystyle \left \lvert \sum_{i=0}^{n-1} g(T^i x) \right \rvert < C d(T^n x, x)^{\alpha}$

we will have shown that the function ${f}$ as defined extends to an ${\alpha}$ -Holder (with constant ${C}$ ) function on ${X}$ .

To see this, we use a stronger form of the closing lemma (which can, I believe, be deduced from the usual one, though I’m not all that solid on the details). Namely, it is that if ${d(T^n x, x)}$ are close, then there exists ${p}$ with ${T^n p =p}$ and

$\displaystyle d(T^i x, T^i x) \leq M d(T^nx, x)\lambda^{-\min(i, n-i)}$

for some ${\lambda>1}$ and constant ${M}$ (depending only on ${T,X}$ ). In other words, the iterates of the point ${p}$ get really close to those of ${x}$ in the middle of the orbit. As a result, the sum ${\sum d(T^i x, T^i p )^{\alpha}}$ is bounded by a constant (depending only on ${T,X}$ !) times ${d(x,p)^{\alpha}}$ . This is a very significant generalization of the usual Anosov closing lemma.

So, if ${d(T^nx, x)}$ is small, we can choose ${p}$ as above, and we have by the vanishing of the periodic obstruction

$\displaystyle \left \lvert \sum_{i=0}^{n-1} g(T^i x) \right \rvert = \left \lvert \sum_{i=0}^{n-1} g(T^i x) - g(T^i p )\right \rvert.$

Now, we have ${d(T^i x, T^i p) \leq M d(T^nx, x)\lambda^{-\min(i, n-i)}}$ , and ${g}$ is ${\alpha}$ -Holder, for some ${K}$ depending on ${g}$ , we have

$\displaystyle \left \lvert \sum_{i=0}^{n-1} g(T^i x) \right \rvert \leq \sum_{i=0}^{n-1} KM^{\alpha} d(T^nx,x)^{\alpha} \lambda^{-\alpha \min(i, n-i)}.$

Because ${\lambda>1}$ , this last expression is bounded by a constant multiple of ${d(T^nx,x)^{\alpha}}$ , whence it follows that ${f}$ extends to ${X}$ as an ${\alpha}$ -Holder function. This completes the proof of Livsic’s theorem.

It is interesting that the result holds if ${\mathbb{R}}$ is replaced by a complete topological group with a bi-invariant metric (e.g. a compact group or an abelian metric group). The proof is essentially the same, and may be left as an exercise for the interested reader. However, for a group like ${GL_n}$ , the proof entirely breaks down. Instead of showing that a sum

$\displaystyle \sum_{i=0}^{n-1} g(T^i x)$

is close to zero, where ${g}$ has vanishing periodic data, one has to show that a product of matrices

$\displaystyle \prod g(T^i x)$

is close to the identity. This is substantially harder because matrices are noncommutative and one cannot simply rewrite this as ${\prod g(T^i x) g(T^i p)^{-1}}$ , for instance. There is, however, such a noncommutative analog of the Livsic theorem, and next time I will talk about it.

Climbing Mount Bourbaki