Ergodic properties of triangle partitions

We study the ergodic properties of a map called the Triangle Sequence. We prove that the algorithm is weakly convergent almost surely, and ergodic. As far as we know, it is the first example of a 2-dimensional algorithm where a surprising diophantine phenomenon happens: there are sequences of nested cells whose intersection is a segment, although no vertex is fixed. Examples of n-dimensional algorithms presenting this behavior were known for n ≥ 3.