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

Abstract

Open conjectures state that, for every x ∈ [0, 1], the orbit (xn) ∞ n=1 of the meanmedian recursion, 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, first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.