Lyapunov exponents and rigidity phenomena in dynamical systems and mathematical physics
This fellowship builds on the success of the applicant’s PhD thesis, where he made breakthroughs in the study of the frequency of hyperbolic behavior, i.e. simplicity and non-vanishing of Lyapunov exponents in dynamical systems. The concept of Lyapunov exponent were introduced in the work of A. M. Lyapunov on the stability of the solutions of differential equations in 1892. The concept plays a central role in most areas of modern theory of dynamical systems, mathematical physics, differential geometry, among others. The problem of the frequency of hyperbolic behavior, in various settings, has been extensively investigated by many leading mathematicians. The main goal of this project is to study the Lyapunov exponents in the most general setting, and investigate its role in rigidity phenomena in dynamical systems. This project specifically aims at applying techniques from modern theories of complex analysis and complex geometry (field in which the supervisor is a world leading expert) to the study of Lyapunov exponents (the applicant’s area of expertise).

This proposal has four main objectives:
1 - Explain the frequency of the simplicity of Lyapunov spectrum for symplectic cocycles;
2 - Establish positivity of Lyapunov exponents for the general structure group;
3 - Understand the frequency of hyperbolic behavior for infinite dimensional ""symplectic"" cocycles;
4 - Employ new techniques from Lyapunov exponents to the rigidity conjectures, in particular, Katok-Spatzier types conjectures.