Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms

Franks, John

doi:10.1090/S0002-9947-96-01502-4

Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
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by John Franks

Trans. Amer. Math. Soc. 348 (1996), 2637-2662

DOI: https://doi.org/10.1090/S0002-9947-96-01502-4

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Abstract:

We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if $0$ is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then $f$ has a fixed point of positive index. The second result asserts that if $f$ has a vanishing mean rotation vector then $f$ has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.

References

Marcy Barge and John Franks, Recurrent sets for planar homeomorphisms, From Topology to Computation: Proceedings of the Smalefest (Berkeley, CA, 1990) Springer, New York, 1993, pp. 186–195. MR 1246118
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133, DOI 10.1090/cbms/038
Andreas Floer, Proof of the Arnol′d conjecture for surfaces and generalizations to certain Kähler manifolds, Duke Math. J. 53 (1986), no. 1, 1–32. MR 835793, DOI 10.1215/S0012-7094-86-05301-9
Martin Flucher, Fixed points of measure preserving torus homeomorphisms, Manuscripta Math. 68 (1990), no. 3, 271–293. MR 1065931, DOI 10.1007/BF02568764
John Franks, Recurrence and fixed points of surface homeomorphisms, Ergodic Theory Dynam. Systems 8$^*$ (1988), no. Charles Conley Memorial Issue, 99–107. MR 967632, DOI 10.1017/S0143385700009366
John Franks, Generalizations of the Poincaré-Birkhoff theorem, Ann. of Math. (2) 128 (1988), no. 1, 139–151. MR 951509, DOI 10.2307/1971464
John Franks, A new proof of the Brouwer plane translation theorem, Ergodic Theory Dynam. Systems 12 (1992), no. 2, 217–226. MR 1176619, DOI 10.1017/S0143385700006702
John Franks, Rotation numbers for area preserving homeomorphisms of the open annulus, Dynamical systems and related topics (Nagoya, 1990) Adv. Ser. Dynam. Systems, vol. 9, World Sci. Publ., River Edge, NJ, 1991, pp. 123–127. MR 1164881
John Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Invent. Math. 108 (1992), no. 2, 403–418. MR 1161099, DOI 10.1007/BF02100612
Shui-Nee Chow, John Mallet-Paret, and James A. Yorke, A periodic orbit index which is a bifurcation invariant, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 109–131. MR 730267, DOI 10.1007/BFb0061414
Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111. MR 979601, DOI 10.2307/1971486
Michael Handel, A Fixed Point Theorem for Planar Homeomorphisms, Preprint.
Michael Handel, Zero Entropy Surface Diffeomorphisms, Preprint.
H. Hadwiger and J. Rätz, Zur Deckungsmonotonie von Inhaltsoperatoren, Math. Nachr. 27 (1963/64), 145–161 (German). MR 165072, DOI 10.1002/mana.19640270303
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942, DOI 10.1515/9781400878055
J. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Annals of Mathematics 42 (1941), 874-920.
Jean-Claude Sikorav, Points fixes d’une application symplectique homologue à l’identité, J. Differential Geom. 22 (1985), no. 1, 49–79 (French). MR 826424