Geometry of affine and autonomous dynamical systems and group actions
Scharif University of Technology
This thesis focuses on the study of “autonomous dynamical systems”. The research revolves around compact smooth manifolds endowed with a parallelization F of their tangent bundle, along with a diffeomorphism of the manifold that possesses a constant derivative cocycle with respect to F. Additionally, the concept of autonomous G-action on the manifold is also introduced, where an action is considered autonomous to mean that any element of it is autonomous.
The first part of the study concentrates on autonomous dynamics on compact 2-manifolds, aiming to classify all autonomous diffeomorphisms on such manifolds. Interestingly, the classification approach presented in dimension two holds promise for extension to higher dimensions.
In the next step, the investigation extends to autonomous dynamics on compact 3-manifolds by considering autonomous partially hyperbolic diffeomorphisms on M. A comprehensive classification is presented, which includes not only the manifold M but also the dynamics for being algebraic. It is shown that, up to finite power and cover, the autonomous dynamics is conjugate to an affine automorphism on a quotient space derived from a simply connected three-dimensional Lie group by a cocompact lattice. The Lie group belongs to one of four possibilities: R^3, the Heisenberg group, Lor(1,1), or the universal cover of SL(2,R).
Furthermore, the regularity of the framing in both two-dimensional and three-dimensional cases is discussed. In three dimensions, the results are extended to the case of framing being C^k, where k is higher or equal to 1. In the case of a C^0 framing, an extension of an Anosov toral diffeomorphism is introduced, which is partially hyperbolic but does not exhibit general algebraic properties. Consequently, a minimum requirement of C^1-autonomous is needed in order to achieve the mentioned classification.
Thesis defense of Mr MOGHADDAMFAR Kambiz under the supervision of Mr ZEGHIB Abdelghani et the co-supervision of Mr FANAI Hamid-Reza