On the Unboundedness of the Transit Time of Mean-Median Orbits

The transit time of mean-median orbits —the time it takes for an orbit to become stationary— has been conjectured to be finite but unbounded over the rationals. Through a study of some near-regular structures in these orbits, we construct two non-trivial sequences of initial sets of increasing size for which the transit time grows linearly and quadratically, respectively, with the size of the set.