Geodesics on \(S^ 2\) and periodic points of annulus homeomorphisms. (English) Zbl 0766.53037

Franks, John

Invent. Math. 108, No. 2, 403-418 (1992).

It is the main result of this paper that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point has infinitely many interior periodic points. It was shown by Birkhoff that closed geodesics on the 2-sphere with a metric of positive Gaussian curvature can be described as periodic points of an area preserving annulus map. Together with recent work by Victor Bangert it follows from the main result of this paper that for every Riemannian metric on the 2- sphere there are infinitely many closed geodesics.

Reviewer: H.-B.Rademacher (Bonn)

MSC:

53C22	Geodesics in global differential geometry
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

area preserving homeomorphism; annulus; periodic point; closed geodesics

References:

[1]	[Ba] Bangert, V.: Geodätische Linien auf Riemannschen Mannigfaltigkeiten. Jahresber. Dtsch. Math. Ver.87, 39–66 (1985) · Zbl 0565.53028
[2]	[Ba2] Bangert, V.: On the existence of closed geodesics on two-spheres (preprint)
[3]	[BH] Bestvina, M., Handel, M.: Personal communication
[4]	[B] Birkhoff, G.D.: Dynamical Systems (Colloq. Publ., Am. Math. Soc., vol. 9) Providence, RI.: Am. Math. Soc. 1927
[5]	[C] Conley, C.: Isolated Invariant Sets and the Morse index. (Reg. Conf. Ser. Math., vol. 38) Providence, RI: Am. Math. Soc. 1978 · Zbl 0397.34056
[6]	[F1] Franks, J.: Recurrence and Fixed Points of Surface Homeomorphisms. Ergodic Theory Dyn. Syst.8 *, 99–107 (1988) · Zbl 0634.58023 · doi:10.1017/S0143385700009366
[7]	[F2] Franks, J.: Generalizations of the Poincaré-Birkhoff Theorem. Ann. Math.128, 139–151 (1988) · Zbl 0676.58037 · doi:10.2307/1971464
[8]	[F3] Franks, J.: A New Proof of the Brouwer Plane Translation Theorem. Ergodic Theory Dyn. Syst. (to appear)
[9]	[F4] Franks, J.: Rotation Numbers for Area Preserving Homeomorphisms of the Open Annulus. (Preprint)
[10]	[Fr] Fried, D.: The Geometry of Cross Sections to Flows. Topology21, 353–371 (1982) · Zbl 0594.58041 · doi:10.1016/0040-9383(82)90017-9
[11]	[G] Grayson, M.: Shortening embedded curves. Ann. Math.129, 71–111 (1989) · Zbl 0686.53036 · doi:10.2307/1971486
[12]	[H1] Handel, M.: Zero Entropy Surface Diffeomorphisms. (Preprint)
[13]	[H2] Handel, M.: There are no Minimal Homeomorphisms of the Multipunctured Plane. (Preprint) · Zbl 0769.58037
[14]	[OU] Oxtoby, J., Ulam, S.: Measure preserving homeomorphisms and metrical transitivity. Ann. Math.42, 874–920 (1941) · Zbl 0063.06074 · doi:10.2307/1968772
[15]	[P] Pollicott, M.: Rotation Sets for Homeomorphisms and Homology. (Preprint) · Zbl 0758.58018
[16]	[S] Schwartzman, S.: Asymptotic Cycles. Ann. Math.66, 270–284 (1957) · Zbl 0207.22603 · doi:10.2307/1969999
[17]	[Th] Thurston, W.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc.19, 417–431 (1988) · Zbl 0674.57008 · doi:10.1090/S0273-0979-1988-15685-6