Ergodic Properties of Smooth Systems on Manifolds and Related Topics

Chaotic properties of smooth dynamical systems is a very active area of research with many open directions and applications in physics and geometry. This project develops a general framework for studying ergodic and statistical properties of such systems. One of the main motivating questions for the project is: to what extent can a deterministic dynamical system resemble a sequence of independent coin tosses. The main tools for studying this question come from ergodic theory, probability and geometry. The work of the project will result in progress in our understanding of fundamental dynamical phenomena with possible consequences and applications in other areas of mathematics, such as geometry and number theory, and also in physics and economics. The project provides research training opportunities for graduate and undergraduate students and postdoctoral researchers. The project is part of an ongoing program of studying ergodic and statistical properties of smooth systems on manifolds and their interactions with geometry and number theory. The Principal Investigator (PI) focuses on the following three main directions: 1. Chaotic properties for systems with non-zero exponents. Ergodic and statistical properties of smooth systems are quite well understood in the case where all Lyapunov exponents are non-zero (hyperbolic systems). On the other hand, ergodic theory of systems for which some (but not all) exponents are zero is much less understood. The PI studies appearance and interactions of ergodic and statistical properties for smooth dynamical systems; 2. Slow chaos for entropy zero systems. The PI plans to investigate quantitative chaotic properties such as rates of mixing, rates of equidistribution, spectral nature and limit theorems for wide classes of zero entropy systems. Even though recent years have seen a lot of progress on these questions, particularly for algebraic (unipotent) examples, there are wide classes of zero entropy systems for which these fundamental properties are not yet well studied; 3. Sparse equidistribution. Recently, there has been a lot of progress on understanding the behavior of orbits of dynamical systems when sampled at sparse subsets of the integers. The PI plans to continue his study of sparse ergodic theorems for wide classes of dynamical systems with emphasis on homogeneous examples where despite the recent progress, many questions are still open. This component of the project will focus on the behavior of orbits along polynomial sequences and also the sequence of k-primes. 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. NSF Award ID: 2554280 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Adam Kanigowski | Institution: University of Maryland, College Park, COLLEGE PARK, MD | Award Amount: $300,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2554280 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2554280.html