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

Given a compact space , a continuous map , and a continuous with vanishing periodic data as in yesterday’s post, there are not many ways to construct a solution of the cohomological equation

The basic thing to note that if is known, then recursively we can determine on the entire orbit of in terms of . In case the map is topologically transitive, say with a dense orbit generated by , then by continuity the entire map is determined by its value on .

This also provides the method for obtaining in the topologically transitive case. Namely, one picks aribtrarily, defines in the only way possible by the cohomological equation. In this way one has defined on the entire orbit such that on this orbit, the equation is satisfied. If one can show that is uinformly continuous on , 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 be a compact Riemannian manifold, a topologically transitive Anosov diffeomorphism. If is an -Holder function such that implies , then there exists an -Holder such that

As already discussed, the proof begins by choosing such that the orbit is dense, and defining

The point is now to prove that if and are close, then are close. Now, the closeness of is the same as saying that is “almost periodic” with period (say for definiteness). The difference is the sum of over the near-orbit

So, if we show that the sum of over a “near orbit” is small, then we will have shown the continuity of . 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 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 such that if for any , then

we will have shown that the function as defined extends to an -Holder (with constant ) function on .

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 are close, then there exists with and

for some and constant (depending only on ). In other words, the iterates of the point get **really** close to those of in the middle of the orbit. As a result, the sum is bounded by a constant (depending **only** on !) times . This is a very significant generalization of the usual Anosov closing lemma.

So, if is small, we can choose as above, and we have by the vanishing of the periodic obstruction

Now, we have , and is -Holder, for some depending on , we have

Because , this last expression is bounded by a constant multiple of , whence it follows that extends to as an -Holder function. This completes the proof of Livsic’s theorem.

It is interesting that the result holds if 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 , the proof entirely breaks down. Instead of showing that a sum

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

is close to the identity. This is substantially harder because matrices are noncommutative and one cannot simply rewrite this as , for instance. There is, however, such a noncommutative analog of the Livsic theorem, and next time I will talk about it.

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