Invariant circles and rotation bands in monotone twist maps (Q1102007)

Fixed points and invariant circles of area-preserving twist maps on an annulus were studied extensively by \textit{G. D. Birkhoff} [Collected Mathematical Papers, Vol. II (Dover, New York) (1968; Zbl 0225.01009)], in view to their applications to celestial mechanics. Said maps are essentially monotone in a finite number of real intervals. The author shows that the existence of certain orbits or minimal sets in an area- preserving monotone twist map of the annulus A is necessary and sufficient for the nonexistence of invariant circles with specified irrational rotation numbers. This generalizes some results of \textit{P. Boyland} and \textit{G. R. Hall} [Topology 26, 21-35 (1987; Zbl 0618.58032)], as well as classical results of Birkhoff and recent results of \textit{J. Mather} [Ergodic Theory Dyn. Syst. 4, 301-309 (1984; Zbl 0557.58019)]. The essential point of the sufficiency depends on the notion of rotation band for an invariant set Z of the map. It is defined as an open interval of real numbers measuring how much f does not preserve the angular order on Z. The main result can be reformulated more precisely as follows: If \(\omega\) is an irrational number between the rotation numbers of f when it is restricted to the boundary circles of A, the conditions for non- existence of an invariant circle with rotation number \(\omega\) are: the existence of an orbit (periodic or not) or a Denjoy minimal set, whose rotation band contains \(\omega\). A Denjoy minimal set is an invariant set of f with some technical conditions which make it topologically behave like some minimal sets in a Denjoy \(C^ 1\) homeomorphism. The relation of these results to other known criteria for nonexistence of invariant sets is not clear. The extension to the case \(\omega\) rational presents more technical difficulties.