Minimal sets of non-resonant torus homeomorphisms

Volume 211 / 2011

Ferry Kwakkel Fundamenta Mathematicae 211 (2011), 41-76 MSC: Primary 37B99; Secondary 37B45. DOI: 10.4064/fm211-1-3


As was known to H. Poincaré, an orientation preserving circle homeomorphism without periodic points is either minimal or has no dense orbits, and every orbit accumulates on the unique minimal set. In the first case the minimal set is the circle, in the latter case a Cantor set. In this paper we study a two-dimensional analogue of this classical result: we classify the minimal sets of non-resonant torus homeomorphisms, that is, torus homeomorphisms isotopic to the identity for which the rotation set is a point with rationally independent irrational coordinates.


  • Ferry KwakkelMathematics Institute
    University of Warwick
    CV4 7AL, Coventry, United Kingdom

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