NSF Award
2418590
Discovered in the 1960s, the Lorenz attractor, and singular flows in general, are a perfect embodiment of a chaotic dynamical system: a deterministic system that, as time evolves, exhibits random and unpredictable behavior. Understanding the statistical structure of such systems has been a central topic in dynamical systems due to their strong chaotic behaviors and deep connections with natural sciences such as statistical physics and atmospheric sciences. However, the existence of singularities makes the system highly resistant to rigorous mathematical analysis from both conceptual and numerical points of view. In this project, the PI aims to develop new tools to study the structures of singular flows in any dimension from the perspective of invariant measures, using recent advances in geometry, probability, and analysis. The educational goal of this project is to develop a pipeline at Wake Forest University for the recruitment, retention, and early-stage research exposure of undergraduate students, graduate students, and postdoctoral scholars, especially those who are traditionally underrepresented in mathematics. The project will support an annual workshop to allow students to disseminate the results of their research and serve as a bridge between mathematicians of diverse backgrounds.<br/><br/>In this project, the PI will develop new tools to study the statistical and topological properties of sectional-hyperbolic and, more generally, multi-singular hyperbolic flows in any dimension, including the classical Lorenz attractor as a special case. The first part of this project focuses on the expansiveness of multi-singular hyperbolic flows, which will lead to the existence of equilibrium states (measures that maximize the measure-theoretic pressure) for continuous potential functions. In the second part of this project, the PI will use the improved Climenhaga-Thompson criterion that he recently developed to show that under mild assumptions, each multi-singular hyperbolic homoclinic class supports a unique equilibrium state. This part requires three main ingredients: Liao's theory on the scaled tubular neighborhood, a fake foliation system on the normal bundle of non-singular orbits, and a recurrence estimate on how typical points return to a small neighborhood of the singular set. As an application, the PI plans to give a partial positive answer to the Spectral Decomposition Conjecture for singular star flows. In the third part of this project, the PI will study the existence of equilibrium states with singular supports and the possibility of having multiple equilibrium states coexisting on the same homoclinic class. For systems with strong hyperbolicity, such examples do not exist in the absence of singularities and, therefore, highlight the fundamental differences between singular and singularity-free systems. Tools developed in the second part of the project have the potential to be applied to other areas of mathematics.<br/><br/>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.
