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Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. (English) Zbl 0893.58037

The aim of this paper is the study of the existence and smoothness of solutions of the cohomological equation \(Xu=f\), for a generic smooth vector field \(X\) on a compact orientable surface \(M\) of genus \(g\geq 2\), which preserves a smooth area form \(\omega\) and has a finite set \(\Sigma\subset M\) of saddle-type singularities and \(f\) a smooth function on \(M\). The three principal theorems are the following. In Theorem A, the author proves that for a “full measured” set of such vector fields, there exists a distributional solution of the cohomological equation; in Theorem B, smooth solutions are studied, and in Theorem C it is proved that the vector space of \(X\)-invariant distributions on \(M\setminus\Sigma\) (for almost all \(X\)) has infinite (countable) dimension.
The paper is divided as follows: 1. Measured foliations and quadratic differentials; 2. Fourier analysis for holomorphic quadratic differentials; 3. A partial isometry associated with a quadratic differential; 4. Existence of solutions and invariant distributions; 5. Applications to smooth area-preserving vector fields.

MSC:

37A99 Ergodic theory
37C10 Dynamics induced by flows and semiflows
58C35 Integration on manifolds; measures on manifolds
46F25 Distributions on infinite-dimensional spaces
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