RENORMALIZATION IN HOLOMORPHIC DYNAMICS AND MLC

Theory of Dynamical systems (with discrete time) studies the long-term behavior of trajectories described by a certain iteration procedure, and the way this phase portrait depends on the parameters of the system. Very interesting fractal objects (like Julia sets and the Mandelbrot set) may appear as phase and parameter diagrams for such systems. The project is focused on complex low-dimensional dynamical systems described by simple quadratic equations. The project will gain deeper insights into small scale structure of dynamical systems. Despite simplicity of the description, these systems are known to display complicated chaotic behavior serving as a good model for various phenomena that appear in celestial mechanics, fluid dynamics, statistical mechanics, biology, and other branches of natural science. The project supports the training postdocs and graduate students who will apply their skills in academia and industry and interactions between senior and junior experts in various branches of real and complex dynamics, and promotes communication in the field through conferences, mini-courses, and a dynamics website (http//www.math.stonybrook/dynamics). The project outlines a program that would lead to a comprehensive topological and geometric understanding of the complex quadratic family. It includes a proof of the MLC Conjecture on the local connectivity of the Mandelbrot set M, which is the central open problem in the field; followed by a unified Renormalization Conjecture for five relevant renormalization schemes that control the small scale structure of M and the corresponding Julia sets in all locations; followed by applications to various topological and geometric properties of Julia sets and the Mandelbrot set. Potential applications include precise topological and geometric models for the Mandelbrot set and Julia sets, full topological classification of neutral quadratic polynomials with arbitrary rotation numbers, and a deeper insight into the nature of Julia sets of positive area. These developments will be reflected in a multi-volume online book “Conformal Geometry and Dynamics of Quadratic Polynomials”. 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: 2555166 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Mikhail Lyubich | Institution: SUNY at Stony Brook, STONY BROOK, NY | Award Amount: $295,477 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2555166 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2555166.html