On Barycentric Subdivision

Abstract

Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that, almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values go to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0, 1] ⊂ ℂ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0, 1/2]. In addition we prove that the largest angle converges to π in probability. Our approach is probabilistic, and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0, 1/2]. The stationary distribution of this limit chain is particularly important in our study.