Cohomological equations occur frequently, sometimes in disguised forms, in dynamical systems. When the dynamics are determined by a diffeomorphism \(f\) on a manifold, the most basic cohomological equation has the form of a first order linear difference equation \[ uf- u=\phi,\tag{\(*\)} \] where \(f: M\to M\) is a diffeomorphism, \(\phi: M\to \mathbb{R}\) is given and \(u\) is unknown. The authors assume that \(f\) and \(\phi\) are \(C^\infty\) and ask for smooth solutions \(u\). By analogy with the cohomology of groups, the function \(\phi\) can be thought of as a smooth cocycle over \(f\) and call it a (smooth) coboundary whenever \((*)\) has a \(C^\infty\) solution. This leads to a natural definition of the first cohomology space \(H^1(f,C^\infty(M))\).
An open question of some interest has been the computation of the \(C^\infty\) first cohomology space of an arbitrary minimal diffeomorphism.
The authors’ main result is the following:
If \(F: \mathbb{T}\to\mathbb{T}\) is an orientation-preserving diffeomorphism of the circle with irrational rotation number, and if \(\mu\) is its only invariant probability measure, then \(\mu\) is the only \(F\)-invariant distribution (up to multiplication by a real constant). A significant corollary is that a minimal \(C^\infty\) circle diffeomorphism is cohomologically \(C^\infty\)-stable if and only if its rotation number is Diophantine.
The authors note that their result is strictly one-dimensional. They provide relevant examples of smooth \(\mathbb{T}^2\) diffeomorphisms that are topologically conjugate to rigid rotations and exhibit higher-order invariant distributions in the upcoming paper titled “Invariant distributions for higher dimensional quasiperiodic maps”.