Ergod. Th. & Dynam. Sys. (1988), 8, 555-584 Printed in Great Britain Regular dependence of invariant curves and Aubry—Mather sets of twist maps of an annulus RAPHAEL DOUADY* Ecole Polytechnique, Centre de Mathematiques, 91128 Palaiseau Cedex, France (Received 29 July 1986 and revised 30 June 1987) Abstract. We prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets. Then we r how that invariant curves of the same map with dif^rent rotation numbers w and <o' cannot approach each other at a distance less than cst. | <w -<w'|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure. Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Poschel (for the second one), in the case of Hamiltonian systems and area preserving maps. 1. Introduction Let / : T 1 x [-S, S] -> T 1 x [-S, S] (0,r)~(e,R), (where T 1 =R/Z) be a diffeomorphism which is isotopic to the identity map and satisfies the monotone twist condition: d@/dr>0. Let o) + (resp. w_) be the rotation number of /| T 'x{s} (resp./| T ' x{ _g } ). Mather [10] and Aubry-Le Daeron [1] (see also Chenciner [3] and Hedlund [4]) proved that, if/ preserves area, then for any o> e [w_, w + ], there exists an orbit off: (0n,r H )=r(e o ,r o ), neZ and a (non-necessarily continuous) map /i^T 1 -»¥' preserving cyclic order and such that: holds for every integer n. When <a is irrational, the closure T m of this orbit is the graph of a Lipschitz map: where * Partially supported by U.A. du CNRS no. 169: MSRI, Berkeley California, USA and IMPA/CNPq, Rio de Janeiro, Brasil. Downloaded from https://www.cambridge.org/core. 18 Jun 2020 at 16:15:14, subject to the Cambridge Core terms of use.