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

Sylvain Crovisier·Francois Beguin·Frederic Le Roux·Alice Patou
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Abstract

Consider a homeomorphism h of the closed annulus S^1*[0,1], isotopic to the identity, such that the rotation set of h is reduced to a single irrational number alpha (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 component of the annulus to the other one, such that gamma is disjoint from its n first iterates under h. As a corollary, we obtain that the rigid rotation of angle alpha can be approximated by homeomorphisms conjugate to h. The first result stated above is an analog of a theorem of J. Kwapisz dealing with diffeomorphisms of the two-torus; we give some new, purely two-dimensional, proofs, that work both for the annulus and for the torus case.

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Sylvain Crovisier, Francois Beguin, Frederic Le Roux, Alice Patou. 2003-09-30. Pseudo-rotations of the closed annulus : variation on a theorem of J. Kwapisz. https://arxiv.org/abs/math/0309477

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