Abstract
This article deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus \(\mathbb {T}^{2}\) which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior. We define two types of points: inessential and essential. The set of inessential points \(\operatorname {Ine}(f)\) is shown to be a disjoint union of periodic topological disks (“elliptic islands”), while the set of essential points \(\operatorname {Ess}(f)\) is an essential continuum, with typically rich dynamics (the “chaotic region”). This generalizes and improves a similar description by Jäger. The key result is boundedness of these “elliptic islands”, which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. We also show that the dynamics in \(\operatorname {Ess}(f)\) is as rich as in \(\mathbb {T}^{2}\) from the rotational viewpoint, and we obtain results relating the existence of large invariant topological disks to the abundance of fixed points.
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Notes
By a theorem of Epstein [6], this is equivalent to saying that f is homotopic to the identity.
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The authors would like to thank T. Jäger and the anonymous referees for the corrections and suggestions that helped improve this paper.
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The first author was partially supported by CNPq-Brasil. The second author was partially supported by FAPESP and CNPq-Brasil.
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Koropecki, A., Tal, F.A. Strictly toral dynamics. Invent. math. 196, 339–381 (2014). https://doi.org/10.1007/s00222-013-0470-3
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DOI: https://doi.org/10.1007/s00222-013-0470-3