Lyapunov Exponents, Rigidity, and Statistics

A central question in dynamical systems is how physical and abstract systems come to exhibit chaotic behavior and how this behavior can be statistically quantified. This project explores chaotic behavior in random dynamical systems, where at every time step the evolution of the system need not be deterministic. Random dynamical systems occur throughout science. For example, many physical processes, such as geophysical systems, are well modeled as random or noisy dynamical systems. Even in areas such as machine learning and AI, random dynamical systems appear: for example, stochastic gradient descent, a popular technique used for training models in machine learning, is a random dynamical system whose long-term iterates produce the trained model. Motivated by such applications, it is useful to understand how quickly random dynamical systems converge to equilibrium and, moreover, what statistical properties that equilibrium state has. This work also provides opportunities for students to conduct research and disseminates these techniques through expository writing and talks. The main goal of this project is the development of a framework that demonstrates the ubiquity of strong chaotic and statistical properties, such as the central limit theorem, in random dynamical systems. Rather than focusing on a particular system, the framework aims to treat conditions that apply broadly across random dynamical systems. It seems that even a very small amount of noise is enough to guarantee strong stochastic properties, which would explain the ubiquity of chaotic behavior. We focus on certain average expansion conditions that appear to be generic, and which imply strong statistical properties. An important feature of these conditions is that they can hold for systems that are far from linear and are computer-verifiable in concrete examples. As chaotic behavior is highly dependent on Lyapunov exponents, which measure the infinitesimal divergence of trajectories, their role is crucial for this goal. Statistical properties can be investigated through a variety of techniques including harmonic analysis, construction of couplings, as well as the use of entropy. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.