The Mean-Median Map

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

“…], 0 ifx ∈ ( 1 2 , 1), where equalities in (0, 1 2 ] occur at unit fractions, whereas those in [ 1 2 , 1) occur at fractions whose numerator and denominator differ by 1. These two families of fractions [10] The Akiyama mean-median map 307 form two sequences, converging to the points 0 and 1 where m A is discontinuous, along which τ A becomes arbitrarily large.…”

“…The author thanks Shigeki Akiyama, who first suggested this variant of the mmm , Franco Vivaldi, through whom the suggestion was communicated and MATRIX, the organiser of the conference which made the communication possible [10].…”

“…Despite intensive research effort [3][4][5][6][7][8][9][10], these terminating conjectures, as well as two additional conjectures to follow, are still open even in the case of the smallest nontrivial initial multisets: those of size three. The fact that the MMM commutes with elementwise affine transformations [4, Section 3] makes the orbit of every such multiset affine-equivalent to that of a univariate initial multiset [0, x, 1], for some real number x ∈ [ 1 2 , 2 3 ] which we call the initial condition.…”

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

“…2 The middle number after sorting if n is odd, the mean of the middle pair otherwise. Despite intensive research effort [9,4,5,3,7,6,8,10], these terminating conjectures, as well as two additional conjectures to follow, are still open even in the case of smallest nontrivial initial sets: those of size three. The fact that the mmm commutes with elementwise affine transformations [4, Section 3] makes the orbit of every such set affine-equivalent to that of a univariate initial set [0, x, 1], for some real number x ∈ 1 2 , 2 3 which we call the initial condition.…”