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The cohomological equation for area-preserving flows on compact surfaces
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## The cohomological equation for area-preserving flows on compact surfaces

### Giovanni Forni

**Abstract.**
We study the equation $Xu=f$ where $X$ belongs to a class of area-
preserving vector fields, having saddle-type singularities, on a
compact orientable surface $M$ of genus $g\geq 2$. For a "full
measure" set of such vector fields we prove the existence, for
any sufficiently smooth complex valued function $f$ in a finite
codimensional subspace, of a finitely differentiable solution $u$.
The loss of derivatives is finite, but the codimension increases as
the differentiability required for the solution increases, so that
there are a countable number of necessary and sufficient conditions
which must be imposed on $f$, in addition to infinite
differentiability, to obtain infinitely differentiable solutions.
This is related to the fact that the "Keane conjecture"
(proved by several authors such as H.Masur, W.Veech, M.Rees,
S.Kerckhoff, M.Boshernitzan), which implies for "almost
all" $X$ the unique ergodicity of the flow generated by $X$ on
the complement of its singularity set, does not extend to
distributions. Indeed, our approach proves that, for "almost
all" $X$, the vector space of invariant distributions not
supported at the singularities has infinite (countable) dimension,
while according to the Keane conjecture the cone of invariant
measures is generated by the invariant area form $\omega$.

*Copyright American Mathematical Society 1996*

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#### Article Info

- ERA Amer. Math. Soc.
**01** (1995), pp. 114-123
- Publisher Identifier: S 1079-6762(95)03005-9
- 1991
*Mathematics Subject Classification*. Primary 58 .
*Key words and phrases*. Cohomological equation, area-preserving flows,
higher genus surfaces.
- Received by the editors July 19, 1995, and, in revised form, December 15, 1995
- Communicated by Svetlana Katok
- Comments

**Giovanni Forni**

Dipartimento di Matematica,
Università di Bologna,
Piazza di Porta S.Donato 5,
40127 Bologna Italy

*Current address:* DPMMS, University of Cambridge, 16 Mill Lane, CB2 1SB Cambridge UK

*E-mail address:* `forni@dpmms.cam.ac.uk `

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