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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.

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.

Reviewer: Raul Ibañez (Bilbão)

### MSC:

37A99 | Ergodic theory |

37C10 | Dynamics induced by flows and semiflows |

58C35 | Integration on manifolds; measures on manifolds |

46F25 | Distributions on infinite-dimensional spaces |