Alladi Ramakrishnan Hall

Geometry and Dynamics of Higher-Dimensional Continued Fractions

Anton Lukyanenko

George Mason University

Classical continued fraction (CF) theory represents numbers as, e.g., pi=3+1/(7+1/(15...)), and has applications throughout number theory, geometry, and dynamics. While real CFs are well-studied, little is known about complex CFs and other higher-dimensional CF algorithms. In this talk, I will start by providing some intuition for the geometry and dynamics of CFs using some interactive visualizations developed by undergraduate students; and then will share new results on convergence, mixing properties, and invariant measures for CF algorithms in R^n and its non-Euclidean analog the Heisenberg group.