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The Dynamics of a Group of Diffeomorphisms of the Sphere and the Linearization Problem

Khodaeian Karim, Amir | 2022

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 55121 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Safari, Mohammad; Nassiri, Meysan
  7. Abstract:
  8. In chis chcsis we consider a work by Dmitry Dolg:opyat and Raphac) Kriko­ rian (on Simulcanoous Lincariwtion of Diffcomorphisms of Sphere) and a recent preprint by Jonathan Dc\Vitt (Simulrn.ncous Lincarization of Diffco­ morphisms of Isotropic Manifolds). Suppasc /1,h,...,J.. arc pcrturbacions of rotations R1, R2 , ...,Rn of the sphere which generate SOd+I (d 2). \\Tc c". plian how it can be shown that all /; can besimn)tancously conjugated to some new rotations when all the Lyapunov cxponcms of the rcndom walk a.,:;.,sociatc to {/;}arc zero. This rcsn1t is applied to obtain stable ergodicity when d is even.The idea is co translate al1 the information co a linear setting and approach chc problem. with tools of PDE and Harmonic Anal­ ysis. This approach is done by applying the Casimir Laplaci.an on certain reprcsentacions of SOd+1. A very important step is to obtain a Fourier like series on the sphere for vector fields, and control their regularity.Using knowledge from harmonic anlysis on homogenoo1L<, vector bundles, ta.rue es­ timates on the norms of some related linear operators can be given in this setting. The proof of the main theorem 1Li,es a KAM argument. In the end
    stable ergodicity is discussed.
  9. Keywords:
  10. Dynamical Systems ; Topological Conjugacy ; Lyapunov Exponent ; Random Dynamical System ; Harmonic Analysis ; KAM Theory ; Casimir Laplacian ; Rotations of the Sphere

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