The periodic orbits of an area preserving twist map (Q1116139)

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 [\textit{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 [\textit{J. Mather}, Commun. Math. Phys. 94, 141-153 (1984; Zbl 0558.58010)]. Using Conley's generalized Morse theory [see e.g. \textit{C. Conley} and \textit{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.