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

TL;DR

This paper investigates the transit time of mean-median orbits, providing constructions that show the time can grow unboundedly, specifically linearly and quadratically, with the initial set size.

Contribution

It introduces new sequences of initial sets demonstrating unbounded transit times, advancing understanding of the conjecture about mean-median orbit behavior.

Findings

Transit time can grow linearly with initial set size.

Transit time can grow quadratically with initial set size.

Constructed explicit sequences illustrating unbounded transit times.

Abstract

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.