Collaborative Research: NSF-BSF: Geometric Properties of Stationary Measures for Smooth Iterated Function Systems

This project belongs to the field of dynamical systems, which is the mathematical study of processes that evolve over time according to fixed rules. These processes often exhibit complicated and chaotic behavior yet display underlying patterns that can be described in terms of stationary measures, that is, certain probabilistic objects that remain stable as the underlying dynamics evolve. The focus here is on stationary measures that arise in nonlinear systems, with particular attention to their geometric and analytic structure. The goal is to understand when such measures are absolutely continuous, when their Fourier transform decays, and how these properties relate to the dynamics that generate them. The work will bring together researchers based in the United States and Israel and will involve the training of graduate students and postdoctoral fellows. A particular emphasis will be placed on maintaining strong collaborative ties between research groups working in dynamics, geometry, and analysis. The main technical goal of this project is to study the regularity and dimension of stationary measures arising from nonlinear actions, such as self-conformal systems and random matrix products. When the maps involved are real analytic and satisfy appropriate separation properties, one expects to be able to compute their dimension in simple terms such as entropy and Lyapunov exponents, and to determine whether they are absolutely continuous. The project aims to establish these properties by studying the behavior of the system under repeated iteration and by using tools that reveal how randomness and geometry interact at different scales. The project will develop via methods from hyperbolic dynamics, harmonic analysis, homogeneous dynamics, spectral theory of transfer operators, and additive combinatorics. These include the use of appropriate disintegrations of measures, spectral gaps for transfer operators under appropriate assumptions, and comparisons between different criteria for separation. The broader aim is to clarify how non-linearity results in stationary measures enjoying rich multiscale structures, which in turn governs their analytic and geometric properties, and to use this understanding to characterize rigidity and regularity phenomena in these systems. 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.