By V. Bangert (auth.), Urs Kirchgraber, Hans-Otto Walther (eds.)
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3) er yields the so-called Birkhoff periodic orbits of minimum type: if x E and a = p/q with (p, q) = 1 then the corresponding orbit (Xi, Yi)iEl of Ii> satisfies (Xi+q, Yi+q) = (Xi + p, Yi). 8) proves the existence of heteroclinic (resp. homoclinic) orbits connecting neighboring BirkhotI periodic orbits of minimum type. For irrational a E (ao, al) we obtain the existence of Mather sets, cf. 6) THEOREM For every irrational 01 E (ao, al) there exists alP-invariant set Mot ~ Sl X (0, 1) with the following properties: (a) Mot is the graph of a Lipschitz function Y;ot: Aot -+ (0, 1) defined on a closed set Aot ~ SI.
3) shows that (H4) holds also for stationary segments if HE C 2 and D2fJlH < O. Hence Xk = Xk and (xt, ... , xl) ~ (Xi, ... , Xj) imply that xt = Xi or xl = Xj. By our assumption on ~ we obtain X = x*, hence (Xi, ... , Xj) ~ (Xi, ... ,Xj) Similarly (Xi, ... , Xj) ~ (Xi, ... , Xj) so that (Xi, ... , Xj) = (Xi, ... , Xj) is minimal. If o E IR\(Q then <"€= ultot since ultot is totally ordered, cf. 1). 8) Suppose XEultot and 00<0<01. ii=fb(xo), Xi = fi (xo). il < Xl < Xl. 8) are disjoint for s = 0 and s = 1.
Let us assume (X_I - x~ d(xI - xi) > and let i: be a minimal geodesic segment from ( - 1, X-I) to (1, xi). 2) it does not intersect c or c* inside ( - 1,1) x IR. A simple topological argument shows that this is not possible. This proves (H4) . Finally we have to show that our assumption's .... (0, s) is a minimal geodesic' is no restriction of generality. 3). We show that there exists a minimal d-geodesic c which is invariant under the translation T(o, I). Then c does not intersect any of its translates T(k,O)C, k ~ 0.