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