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By V. Bangert (auth.), Urs Kirchgraber, Hans-Otto Walther (eds.)

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2) is true. H is C 2 in the interior of D and D1H, D2H and D~IH extend continuously to D. 1)(b) implies D2D 1H= -D2a ~ -0 for some 0> 0. e. w is even exact on the cylinder 'D modulo 36 Mather Sets for Twist Maps and Geodesics on Tori- V. Bangert translations Tu, j), j E 7L' : w vanishes along the curve ~ -+ (~,J(~, 0» from (O,J(O,O» to (1,J(0,0) + 1). 3), the extended H: fR2 -+ fR satisfies (Hr)-(H4) from Section 1. 2) as definition we can extend ;p resp. tp to fR2 resp. 2) we have a variational principle for the orbits of the extended ;P.

Briefly one can say that uft~ec is either a foliation or a lamination of /R 2 • For a (necessarily sketchy) picture of a lamination see Fig. 6. r for a E (Q U {00 I. No two elements of the same family intersect while two elements of different families intersect exactly once. At~ec corresponds to a Cantor set for all a E /R\(Q. 15) and the corresponding property for circle maps. So we find many examples of complicated minimal sets for a smooth geodesic flow of a torus. The construction of such minimal sets for geodesic flows of surfaces of higher genus is one of the main results in [42].

Ita ;C 0. ita. For every x E vNa \vN~ec there exist ~ < x < x and x and x are asymptotic. 6) ~ E vN~ec, X E vN~ec such that Inasmuch as this is possible Fig. 6 gives a picture of vita in case C( E IR\~ and Po(vlta);c IR. The dotted line represents an xEvlta\vIt~ec. One deficiency in this qualitative description of vNa is that we do not say much about the size of vNa \vN~ec. One expects that for generic H one will have vN~ec = vNa for most a E IR\~, cf. the end of Section 9. At this point we make some brief remarks on the invariant curve problem mentioned in the introduction.

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