Strictly Toral Dynamics

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Talebi, Amin Sadat | 2015

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 47046 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Esfahanizadeh, Mostafa; Nassiri, Meysam
Abstract:
This dissertation deals with nonwandering (e.g. area-preserving) homeomorphisms of the torus 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. There are two types of points: inessential and essential. It is known that the set of inessential points Ine(f ) is a disjoint union of periodic topological disks ("elliptic islands"), while the set of essential points Ess(f ) is an essential continuum, with typically rich dynamics ("the chaotic region"). One of the key results in this context is boundedness of these "elliptic islands", which allows, among other things, to obtain sharp (uniform) bounds of the diffusion rates. It is also known that the dynamics in Ess(f ) is as rich as in from the rotational viewpoint, and there are some results relating the existence of large invariant topological disks to the abundance of fixed points
Keywords:
Strictly Torol Dynamics ; Rotation Set ; Elliptic Islands ; Gradient Like Foliation

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