By V. Bangert (auth.), Urs Kirchgraber, Hans-Otto Walther (eds.)
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One needs an easy estimate to ensure the existence of (q. p )-periodic sequences X E IR z such that (Xi. Xi + I) E D for all i E7L We add a remark due to J. Moser which gives a less technical relation between monotone twist maps and the variational problems of Section 6. The essence is that the relation between a monotone twist map and its generating function is a discrete version of the Legendre transformation. Look at the non- 0». 7 Applications to monotone twist maps 41 parametric version L of a Finsler metric F on /R 2 : L (I, x, x):= F((t, x), (1, x» Then curves (I, x(t)) which are extremals of JL(I, X(I),X(I» dl become F-geodesics if they are reparametrized by arc-length.
Xi*) + H(Xi*' ... , Xi"t- q) - H(Xi"t-q-I. xt+q) + H(Xi+q-I. * Xi + p) + H(Xi, ... , Xj-J) + H(Xj_1 + p, Xj+q) Since X and x* cross between i - 1 and i and since x* is periodic with (q, p) we have H(Xi-l, Xi) > H(Xi-l, xt) + H(Xi*-- I. 5) H(Xi-I. ,Xj+q) = H(Xi-I. , Xj-J) + H(Xi*' ... , Xi*+q) + H(xj-I + p, Xj+q) - e for some e> O. 6) j-co and lim IH(xj, ... , Xj+q) - H(X6, ... 1) we have H(X6, ... , X;) = H(xt, ... , Xi"t- q ), hence lim IH(xj, ... , Xj+q) - H(Xi*' ... 4) holds for large j. e.
2, g) with two disjoint big bumps HI and H2. Let C I and C2 be two disks in the centers of HI and H2. For convenience we assume that C I and C2 are bounded by closed geodesics. Fix an arbitrary irrational 0: E IR. We are going to show that there exists a continuum of disjoint closed sets ~.. , e of geodesics which have exactly the same properties as vI( .. except that they do not consist of minimal geodesics. So this is counter-example to the-at first sight tempting-conjecture that vi( VI can be characterized by such properties.