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ON ANNULAR MAPS OF THE TORUS AND SUBLINEAR DIFFUSION

Part of: Topological dynamics Low-dimensional dynamical systems

Published online by Cambridge University Press:  23 June 2016

Pablo Dávalos
Pablo Dávalos*
Affiliation:
Instituto Tecnológico y de Estudios Superiores de Occidente, Periférico Sur Manuel Gómez Morín 8585, C.P. 45604, Tlaquepaque, Jalisco, México (davalo@gmail.com)
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Abstract

A classical article by Misiurewicz and Ziemian (J. Lond. Math. Soc.40(2) (1989), 490–506) classifies the elements in Homeo$_{0}(\mathbf{T}^{2})$ by their rotation set $\unicode[STIX]{x1D70C}$, according to wether $\unicode[STIX]{x1D70C}$ is a point, a segment or a set with nonempty interior. A recent classification of nonwandering elements in Homeo$_{0}(\mathbf{T}^{2})$ by Koropecki and Tal was given in (Invent. Math.196 (2014), 339–381), according to the intrinsic underlying ambient space where the dynamics takes place: planar, annular and strictly toral maps. We study the link between these two classifications, showing that, even abroad the nonwandering setting, annular maps are characterized by rotation sets which are rational segments. Also, we obtain information on the sublinear diffusion of orbits in the—not very well understood—case that $\unicode[STIX]{x1D70C}$ has nonempty interior.

Keywords

torus homeomorphismsrotation settopological dynamics

MSC classification

Primary: 37B99: None of the above, but in this section 37E45: Rotation numbers and vectors 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 4 , September 2018 , pp. 913 - 978
Copyright
© Cambridge University Press 2016 

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Footnotes

This work was partially supported by FAPESP-Brasil grant 2011/14122-5 and ITESO-México.

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