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Pseudo-rotations of the closed annulus : variation on a theorem of J. Kwapisz
François BéguinF. Le RouxA PatouSylvain Crovisiersubject
Mathematics::Dynamical Systems[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS][ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]General Physics and AstronomyBoundary (topology)Dynamical Systems (math.DS)Disjoint sets01 natural sciences37E45 37E30CombinatoricsInteger0103 physical sciencesFOS: Mathematics0101 mathematicsMathematics - Dynamical SystemsMathematical PhysicsMathematicsApplied Mathematics010102 general mathematicsStatistical and Nonlinear PhysicsAnnulus (mathematics)TorusMathematics::Geometric TopologyHomeomorphismIterated function010307 mathematical physicsDiffeomorphismdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2003-09-29 |