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Lyapunov exponents and rigidity phenomena in dynamical systems and mathematical physics

Periodic Reporting for period 1 - LYP-RIG (Lyapunov exponents and rigidity phenomena in dynamical systems and mathematical physics)

Reporting period: 2019-08-07 to 2021-08-06

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.
During the fellowship, researcher studied the Lyapunov exponents of cocycles and get much better understanding on the theory, some partial results, especially the relationship between Lyapunov exponents and the theory of normal forms are obtained, (Project A.1 A.2 Project B and the researcher with his coauthors got surprising applications of his studies in Project D) and certain global picture of theory is still working on progress (that is quite usual in the studies of pure mathematics that some deep results, especially theories may even need more than 5 or 10 years to complete, but luckily the researcher already got applications of his result before completing his plan, more details are given in the following paragraphs).
Moreover the researcher build a very nice connection between the Project A and Project D, as a consequence, the researcher successfully completed the study of some rigidity phenomenon and fully explained the centralizer problem for typical diffeomorphisms of compact manifolds. These works are related to projects A, B and D and appeared in Work Packages WP 1, 2, 4, and have led to two major articles, which were completely written during the period the researcher was employed on the fellowship. Due to the pandemic, and terminating the project during to start his new position, the researcher has no enough time to work on Project C during the fellowship.
Lyapunov exponents