Geometric and Probabilistic Structures in Dynamics by Burns K., Dolgopyat D., Pesin Y. (eds.)

By Burns K., Dolgopyat D., Pesin Y. (eds.)

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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. We recall that a sequence (Xi)iEl is stationary with respect to H if D2H(Xi-1o Xi) + DIH(Xi, Xi+ r) = 0 for all i E 7L, cf.

10) THEOREM Suppose a is irrational. Either there is acE JI ~ec through every point of 1R2 (c is unique up to parametrization) or JI~ec intersects 34 Mather Sets jar Twist Maps and Geodesics on Tori- V. Bangert every periodic minimal geodesic in a Cantor set. At ~ec. At~ec can be approximated by periodic minimal geodesics. 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.

3) tells us that there exists i E Z such that c(s) = (Hs), 7J(s» is contained in [i, i + 1) x fR. We are now in the position to interpret the results of Sections 3-5 as statements on minimal a-geodesics where is a lift of a metric d on fR2/Z2 satisfying (Gd and (G2). In particular all of the following statements are true for symmetric Finsler metrics. Note that the statements are invariant under coordinate changes tl>: fR2 -+ fR2 satisfying tl>(p + k) = tl>(p) + k for all pE fR2,kEZ2. 5) THEOREM Let c(s) = (Hs), 7J(s» be a minimal geodesic in (fR2, a).

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