The Akiyama Mean-Median Map Has Unbounded Transit Time and Discontinuous Limit

TL;DR

This paper investigates a modified mean-median recursion, proving that its transit time is unbounded while its limit function is discontinuous, contrasting with previous conjectures about the original system.

Contribution

It establishes that the modified Akiyama mean-median map has unbounded transit time and a discontinuous limit function, providing new insights into its dynamical behavior.

Findings

Transit time function is unbounded.

Limit function is discontinuous.

Contrasts with previous conjectures about the original map.

Abstract

Open conjectures state that, for every $x\in[0,1]$, the orbit $\left(x_n\right)_{n=1}^\infty$ of the mean-median recursion $$x_{n+1}=(n+1)\cdot\mathrm{median}\left(x_1,\ldots,x_{n}\right)-\left(x_1+\cdots+x_n\right),\quad n\geqslant 3,$$ with initial data $\left(x_1,x_2,x_3\right)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of $x$) are unbounded and continuous, respectively. In this paper we prove that, for the slightly modified recursion $$x_{n+1}=n\cdot\mathrm{median}\left(x_1,\ldots,x_{n}\right)-\left(x_1+\cdots+x_n\right),\quad n\geqslant 3,$$ first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.