New Directions in Thermodynamic Formalism for Geodesic Flows Beyond the Closed Riemannian Case

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A fundamental question in dynamical systems, systems that model natural phenomena changing with time, is to understand their asymptotic behavior. That is, given knowledge of the present, what can we say about the distant future or distant past? In most situations, the answer must be given in terms of probability. This leads to the question of identifying and studying natural (invariant) probability measures. Thermodynamic formalism, which is a dynamical theory originally inspired by statistical mechanics, is a framework for answering this kind of question. The geodesic flow is the dynamical system given by moving at unit speed along paths that minimize distance. This flow has special importance because of its relationship with the geometry and topology of the underlying space. The geodesic flow has inspired many important developments in dynamical systems theory, in particular leading to the definitions on which hyperbolic dynamics is based. This research project pursues a distinctive vision for progress in this area, with focus on developing novel techniques suitable for application to geodesic flows in more general settings. The award also supports the training of graduate and undergraduate students.The project has four parts. Part 1 develops fundamental results in thermodynamic formalism suitable for applications to dynamical systems of geometric origin. The focus is on the non-compact world, building on previous advances made for closed non-positive curvature manifolds. Areas of interest include non-compact CAT(-1) spaces, non-positive curvature manifolds with cusps, and as a long-term goal, thermodynamics for the Weil-Petersson geodesic flow. Part 2 considers statistical and dynamical properties for equilibrium states, particularly Central Limit Theorems and second order differentiability. Part 3 develops a deeper dynamical understanding of geodesic flow for CAT(−1) spaces. We investigate analogues of the SRB measure, particularly in the setting of CAT(−1) metrics on closed surfaces. Part 4 concerns questions about Katok entropy rigidity in dynamical systems, particularly for non-compact surfaces.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.