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The periodic orbits of an area preserving twist map. (English) Zbl 0665.58034
The author studies the oscillation properties of periodic orbits of an area preserving diffeomorphism of the cylinder $$S^ 1\times {\mathbb{R}}$$ onto itself which satisfies an infinite twist condition and also a no flux condition. Periodic orbits of type (p,q) are identified as critical points of an appropriate function W. Two concepts are introduced: (i) the intersection number I(x,y) of two orbits x and y which intersect transversally; this intersection number is nonincreasing under the gradient flow of W [J. Smillie, SIAM J. Math. Anal. 15, 530-534 (1984; Zbl 0546.34007)]; (ii) the twist number $$\tau$$ (x) of an orbit, related to the Morse index of the corresponding critical point of W [J. Mather, Commun. Math. Phys. 94, 141-153 (1984; Zbl 0558.58010)].
Using Conley’s generalized Morse theory [see e.g. C. Conley and E. Zehnder, Commun. Pure Appl. Math. 37, 207-253 (1984; Zbl 0559.58019)] it is then shown that the existence of a periodic orbit with $$\tau (x)>0$$ implies the existence of two other periodic orbits with twist number zero, and for each integer k with $$0<k<\tau (x)$$ the existence of at least two period orbits $$y_ i$$ $$(i=1,2)$$ with $$I(y_ i,x)=2k$$. Finally it is shown that there must exist also some homoclinic or heteroclinic orbits.
Reviewer: A.Vanderbauwhede

MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems 37A99 Ergodic theory 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:
 [1] Arnold, V.I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1978 · Zbl 0386.70001 [2] Aubry, S., le Daeron, P.Y.: The discrete Frenkel-Kontorova model and its extensions. Physica8, 381-422 (1983) · Zbl 1237.37059 [3] Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955 · Zbl 0064.33002 [4] Conley, C.: Isolated invariant sets and the Morse index. C.B.M.S. Reg. Conf. Ser. Math.38, Published by the AMS (1978) · Zbl 0397.34056 [5] Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math.73, 33-49 (1983) · Zbl 0516.58017 · doi:10.1007/BF01393824 [6] Conley, C., Zehnder, E.: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math.37, 207-253 (1984) · Zbl 0559.58019 · doi:10.1002/cpa.3160370204 [7] Dancer, E.N.: Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math.350, 1-22 (1984) · Zbl 0525.58012 · doi:10.1515/crll.1984.350.1 [8] Hirsch, M.: Systems of differential equations which are competitive or cooperative, I: Limit sets. SIAM J. Math. Anal.13, 167-179 (1982) · Zbl 0494.34017 · doi:10.1137/0513013 [9] Matano, H.: Nonincrease of the lap-number of a solution for a one dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo29, 401-441 (1982) · Zbl 0496.35011 [10] Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology21, 457-467 (1982) · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 [11] Mather, J.N.: Amount of rotation about a point and the Morse index. Commun. Math. Phys.94, 141-153 (1984) · Zbl 0558.58010 · doi:10.1007/BF01209299 [12] Moerbeke, P. van: The spectrum of Jacobi matrices. Invent. Math.37, 45-81 (1976) · Zbl 0361.15010 · doi:10.1007/BF01418827 [13] Smillie, J.: Competitive and cooperative tridiagonal systems of differential equations. SIAM J. Math. Anal.15, 531-534 (1984) · Zbl 0546.34007 · doi:10.1137/0515040
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