Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
Author:
John Franks
Journal:
Trans. Amer. Math. Soc. 348 (1996), 26372662
MSC (1991):
Primary 58C30; Secondary 58F11
MathSciNet review:
1325916
Fulltext PDF Free Access
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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 is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then has a fixed point of positive index. The second result asserts that if has a vanishing mean rotation vector then 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.
 [BF]
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
(94m:54095)
 [C]
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
(80c:58009)
 [Flo]
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
(87h:58188), http://dx.doi.org/10.1215/S0012709486053019
 [Flu]
Martin
Flucher, Fixed points of measure preserving torus
homeomorphisms, Manuscripta Math. 68 (1990),
no. 3, 271–293. MR 1065931
(91j:58129), http://dx.doi.org/10.1007/BF02568764
 [F1]
John
Franks, Recurrence and fixed points of surface homeomorphisms,
Ergodic Theory Dynam. Systems 8* (1988), no. Charles
Conley Memorial Issue, 99–107. MR 967632
(90d:58124), http://dx.doi.org/10.1017/S0143385700009366
 [F2]
John
Franks, Generalizations of the PoincaréBirkhoff
theorem, Ann. of Math. (2) 128 (1988), no. 1,
139–151. MR
951509 (89m:54052), http://dx.doi.org/10.2307/1971464
 [F3]
John
Franks, A new proof of the Brouwer plane translation theorem,
Ergodic Theory Dynam. Systems 12 (1992), no. 2,
217–226. MR 1176619
(93m:58059), http://dx.doi.org/10.1017/S0143385700006702
 [F4]
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
(93e:58153)
 [F5]
John
Franks, Geodesics on 𝑆² and periodic points of annulus
homeomorphisms, Invent. Math. 108 (1992), no. 2,
403–418. MR 1161099
(93f:58192), http://dx.doi.org/10.1007/BF02100612
 [CMY]
ShuiNee
Chow, John
MalletParet, 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
(85d:58058), http://dx.doi.org/10.1007/BFb0061414
 [G]
Matthew
A. Grayson, Shortening embedded curves, Ann. of Math. (2)
129 (1989), no. 1, 71–111. MR 979601
(90a:53050), http://dx.doi.org/10.2307/1971486
 [H1]
Michael Handel, A Fixed Point Theorem for Planar Homeomorphisms, Preprint.
 [H2]
Michael Handel, Zero Entropy Surface Diffeomorphisms, Preprint.
 [HDK]
H.
Hadwiger and J.
Rätz, Zur Deckungsmonotonie von Inhaltsoperatoren, Math.
Nachr. 27 (1963/1964), 145–161 (German). MR 0165072
(29 #2363)
 [M]
John
Milnor, Lectures on the ℎcobordism theorem, Notes by
L. Siebenmann and J. Sondow, Princeton University Press, Princeton, N.J.,
1965. MR
0190942 (32 #8352)
 [OU]
J. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Annals of Mathematics 42 (1941), 874920.
 [S]
JeanClaude
Sikorav, Points fixes d’une application symplectique
homologue à l’identité, J. Differential Geom.
22 (1985), no. 1, 49–79 (French). MR 826424
(88g:58066)
 [BF]
 M. Barge and J. Franks, Recurrent Sets for Planar Homeomorphisms, in, From Topology to Computation: Proceedings of the Smalefest (M.W. Hirsch, J.E. Marsden, M. Shub, eds.), SpringerVerlag, 1993, pp. 186195. MR 94m:54095
 [C]
 C. Conley, Isolated Invariant Sets and the Morse index,, vol. 38, C.B.M.S. Regional Conference Series in Math. Amer. Math. Soc., Providence, RI, 1978. MR 80c:58009
 [Flo]
 A. Floer, Proof of the Arnold Conjecture for surfaces and generalizations to certain Kähler manifolds, Duke Math. Jour. 51 (1986), 132. MR 87h:58188
 [Flu]
 Martin Flucher, Fixed points of measure preserving torus homeomorphisms, Manuscripta Mathematica 68 (1990), 271293. MR 91j:58129
 [F1]
 J. Franks, Recurrence and Fixed Points of Surface Homeomorphisms, Ergodic Theory and Dyn. Systems 8 (1988), 99107. MR 90d:58124
 [F2]
 J. Franks, Generalizations of the PoincaréBirkhoff Theorem, Annals of Math. 128 (1988), 139151. MR 89m:54052
 [F3]
 J. Franks, A New Proof of the Brouwer Plane Translation Theorem, Ergodic Theory and Dynamical Systems 12 (1992), 217226. MR 93m:58059
 [F4]
 J. Franks, Rotation Numbers for Area Preserving Homeomorphisms of the Open Annulus, World Science, 1991. MR 93e:58153
 [F5]
 J. Franks, Geodesics on and Periodic Points of Annulus Homeomorphisms, Inventiones Math. 108 (1992), 403418. MR 93f:58192
 [CMY]
 S.N. Chow, J. M. MalletParet, and J. A. Yorke, A periodic orbit index which is a bifurcation invariant, vol. 1007, Springer Lecture Notes in Math., 1983, pp. 109131. MR 85d:58058
 [G]
 Matthew Grayson, Shortening embedded curves, Annals of Math. 129 (1989), 71111. MR 90a:53050
 [H1]
 Michael Handel, A Fixed Point Theorem for Planar Homeomorphisms, Preprint.
 [H2]
 Michael Handel, Zero Entropy Surface Diffeomorphisms, Preprint.
 [HDK]
 H. Hadwiger, H. Debrunner, and V. Klee, Combinatorial Geometry in the Plane, Holt Rinehart and Winston, New York, 1964. MR 29:2363
 [M]
 John Milnor, Lectures on the hcobordism theorem, Princeton Univ. Press, 1965. MR 32:8352
 [OU]
 J. Oxtoby and S. Ulam, Measure preserving homeomorphisms and metrical transitivity, Annals of Mathematics 42 (1941), 874920.
 [S]
 J.C. Sikorav, Points fixes d'une application symplectique homologue à l'identité, Jour. Diff. Geom. 22 (1985), 4979. MR 88g:58066
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Additional Information
John Franks
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 602082730
Email:
john@math.nwu.edu
DOI:
http://dx.doi.org/10.1090/S0002994796015024
PII:
S 00029947(96)015024
Received by editor(s):
September 20, 1994
Received by editor(s) in revised form:
March 31, 1995
Article copyright:
© Copyright 1996
American Mathematical Society
