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

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 $$x_{n+1}=(n+1)\cdot \mathrm{median}\left(x_1,\dots ,x_n\right)-\left(x_1+\cdots +x_n\right),n⩾3,$$ 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 $$x_{n+1}=n\cdot \mathrm{median}\left(x_1,\dots ,x_n\right)-\left(x_1+\cdots +x_n\right),n⩾3,$$ first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

LaTeX, 9 pages with 4 figures