Ergodicity of the Triangle Map in R^n by Jacob Lehmann Duke ’24

Ergodicity of the Triangle Map in R^n by Jacob Lehmann Duke ’24, Monday May 6, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Thesis

Abstract: The study of multidimensional continued fractions is rich and complicated. We examine the geometry of one popular multidimensional continued fraction algorithm, the Triangle Map, deriving it from its one-dimensional counterparts the Gauss and Farey maps. Our main result proves that in any dimension n, this map is ergodic, a desirable mixing property that helps us to better understand its behavior. This resolves a 2009 conjecture of Messaoudi, Noguiera, and Schweiger. We also present several related results, including weak convergence properties, ergodicity of the Slow Triangle Map, normalizing constants, and invariant measures. The talk will focus on the key elements of the proof and the ways in which the general case differs from the n=3 case, which was already known.