During the Project, the researcher completed two joint work on Lyapunov exponents and rigidity phenomena. The first is a joint work with D. Damjanovic and A. Wilkinson, with the co-authors the researcher discovered a rigidity phenomenon within the volume-preserving partially hyperbolic diffeomorphisms with 1-dimensional center. The second is a joint work with S. Gan, Y. Shi, J. Zhang, with the co-authors the researcher showed some very interesting rigidity phenomena for DA maps on three-dimensional torus. Both of these two results are submitted to two prestigious journals for peer reviewing. Besides the researcher and the host have some work in progress for the discussion of centralizer for algebraic defined cocycle over some one-dimensional complex dynamical systems.
During the fellowship, the researcher was the lead organizer of the weekly seminars on dynamical systems at Imperial College. He also presented several talks at intentional conference and research seminars. In addition, researcher organized, and delivered several lectures, at the host institution; and supervised a group project for undergraduate students.
As the researcher obtained a permanent job at a prestigious university in his home country (Peking University) during the fellowship, he ended up terminating the contracted earlier than expected. He was able to obtain partial extension, which allowed him to complete about half of my proposed project. Because the main purpose of this fellowship was to enhance the CV of the researcher to obtain such a permanent role, we consider the fellowship highly successful, and the researcher is grateful for the support from the MC programme to make this possible.
1.1 Objectives
The object of our object is to study Lyapunov exponents and rigidity phenomena in dynamical systems and mathematical physics. More precisely, the researcher planned to deeply investigate the hyperbolic behaviour (or in other words, positivity of Lyapunov exponents) of certain cocycles, and hope to get new understanding for some problems in dynamical systems and rigidity problem of higher rank group actions. The complete proposal consisted of two aspects: conjectures for Lyapunov exponents, and rigidity phenomenon for higher rank group action. The conjectures were listed under the headings "Projects A, B, C” and appeared in Work Packages WP1, 2, 3. The rigidity phenomena was listed under the heading Project D, and appeared in WP4.
