Pseudo-rotations of the closed annulus:: variation on a theorem of J!Kwapisz

Consider a homeomorphism h of the closed annulus S(1) x [0, 1], isotopic to the identity, such that the rotation set of h is reduced to a single irrational number a (we say that h is an irrational pseudo-rotation). For every positive integer n, we prove that there exists a simple arc gamma joining one of the boundary components of the annulus to the other, such that gamma is disjoint from its n first iterates under h. As a corollary, we obtain that the rigid rotation of angle a can be approximated by homeomorphisms that are conjugate to h. The first result stated above is an analogue of a theorem of Kwapisz dealing with diffeomorphisms of the two-torus; we give some new, purely two-dimensional, proofs of this theorem, that work both for the annulus and for the torus case.