The cohomological equation for area-preserving flows on compact surfaces

Forni, Giovanni^1, 2

¹ Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.Donato 5, 40127 Bologna Italy
² DPMMS, University of Cambridge, 16 Mill Lane, CB2 1SB Cambridge UK

Electronic research announcements of the American Mathematical Society, Tome 01 (1995) no. 3, pp. 114-123.

Résumé

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 that 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 $\omega$.

Détail

DOI : 10.1090/S1079-6762-95-03005-8

Bibliographie

[1] Boshernitzan, Michael A condition for minimal interval exchange maps to be uniquely ergodic Duke Math. J. 1985 723 752

[2] Séminaire Bourbaki. Vol. 1984/85 1986 1 373

[3] Chavel, Isaac Eigenvalues in Riemannian geometry 1984

[4] Cheeger, Jeff On the spectral geometry of spaces with cone-like singularities Proc. Nat. Acad. Sci. U.S.A. 1979 2103 2106

[5] Collet, P., Epstein, H., Gallavotti, G. Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties Comm. Math. Phys. 1984 61 112

[6] Travaux de Thurston sur les surfaces 1979 284

[7] Guillemin, V., Kazhdan, D. On the cohomology of certain dynamical systems Topology 1980 291 299

[8] Guillemin, Victor, Kazhdan, David Some inverse spectral results for negatively curved 𝑛-manifolds 1980 153 180

[9] Hubbard, John, Masur, Howard Quadratic differentials and foliations Acta Math. 1979 221 274

[10] Katok, A. B. Invariant measures of flows on orientable surfaces Dokl. Akad. Nauk SSSR 1973 775 778

[11] Keane, Michael Interval exchange transformations Math. Z. 1975 25 31

[12] Keane, Michael Non-ergodic interval exchange transformations Israel J. Math. 1977 188 196

[13] Kerckhoff, S. P. Simplicial systems for interval exchange maps and measured foliations Ergodic Theory Dynam. Systems 1985 257 271

[14] Kerckhoff, Steven, Masur, Howard, Smillie, John Ergodicity of billiard flows and quadratic differentials Ann. of Math. (2) 1986 293 311

[15] Nakayama, Tadasi On Frobeniusean algebras. I Ann. of Math. (2) 1939 611 633

[16] Nakayama, Tadasi On Frobeniusean algebras. I Ann. of Math. (2) 1939 611 633

[17] Livšic, A. N. Certain properties of the homology of 𝑌-systems Mat. Zametki 1971 555 564

[18] De La Llave, R., Marco, J. M., Moriyón, R. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation Ann. of Math. (2) 1986 537 611

[19] Masur, Howard Interval exchange transformations and measured foliations Ann. of Math. (2) 1982 169 200

[20] Moser, J. On invariant curves of area-preserving mappings of an annulus Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962 1 20

[21] Rees, Mary An alternative approach to the ergodic theory of measured foliations on surfaces Ergodic Theory Dynam. Systems 1981

[22] Rudin, Walter Real and complex analysis 1987

[23] Corliss, J. J. Upper limits to the real roots of a real algebraic equation Amer. Math. Monthly 1939 334 338

[24] Dienes, Z. P. Canonic elements in the higher classes of Borel sets J. London Math. Soc. 1939 169 175

[25] Stein, Elias M., Weiss, Guido Introduction to Fourier analysis on Euclidean spaces 1971

[26] Strebel, Kurt Quadratic differentials 1984

[27] Thurston, William P. On the geometry and dynamics of diffeomorphisms of surfaces Bull. Amer. Math. Soc. (N.S.) 1988 417 431

[28] Veech, William A. Projective Swiss cheeses and uniquely ergodic interval exchange transformations 1981 113 193

[29] Zygmund, A. Trigonometric series. 2nd ed. Vols. I, II 1959