## Cohomological equations and invariant distributions for minimal circle diffeomorphisms.(English)Zbl 1225.37052

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”.

### MSC:

 37E10 Dynamical systems involving maps of the circle 37C55 Periodic and quasi-periodic flows and diffeomorphisms 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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### References:

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