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

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