Complex dynamics: group actions, Migdal-Kadanoff renormalization, and ergodic theory

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Dynamical systems is the area of mathematics that studies how the state of a system changes with time. Such systems are abundant in all areas of science including biology, chemistry, and physics. They are also readily visible in our everyday lives, ranging from describing the ways in which a disease epidemic is likely to progress to predicting the weather. Given the initial state of the system, one would like to know what the future state of the system will be, as well as the long-term behavior of the system. The equations describing such real-world phenomena are very complicated and are usually custom tailored to the system at hand. They are typically far too difficult for rigorous study and scientists must often use numerical simulations to analyze them. However, the underlying dynamical phenomena can often be understood by studying simpler systems whose states can be described in terms of one or two variables. This project will support the study of dynamical systems consisting of iterating rational mappings in two complex variables, a setting where the powerful tools of complex analysis and algebraic geometry are available. The research is designed around three principles: (1) exploring connections between two or more different areas of mathematics can lead to surprising new results, (2) dynamical systems having an additional context from another field can be studied significantly more deeply, and (3) a study of concrete examples often leads to more general theories. Among other things, this project will support Ph.D. students from Indiana University-Purdue University Indianapolis to engage in these research topics, thus training them in dynamical systems. The broader impacts of this grant will be further achieved through the principal investigator's mentoring of highly talented high-school students from the Indianapolis area, and running the IUPUI High School Math Contest which engages approximately 60 to 100 high-school students from Indiana each year.This research project is concerned with complex dynamics in higher dimensions. The main goal is to study the iterates of holomorphic (or rational) self-mappings of a complex manifold of dimension two or larger, and, more generally, to study the actions of finitely generated groups of biholomorphic (or birational) self-mappings. The topics to be investigated, which draw connections with other areas of mathematics, include: (1) holomorphic group actions on complex surfaces coming from the monodromy of the Painleve 6 differential equation, (2) Migdal-Kadanoff renormalization mappings associated to phenomena in statistical physics on hierarchical lattices, and (3) ergodic theory of rational maps with transcendental first dynamical degree. Understanding the underlying systems will lead to valuable theoretical results in holomorphic dynamics.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.