On two conjectures for M&m sequences

Abstract

In this paper, the recently introduced M&m sequences and associated mean-median map are studied. These sequences are built by adding new points to a set of real numbers by balancing the mean of the new set with the median of the original. This process, although seemingly simple, gives rise to complicated dynamics. The main result is that two conjectures put forward by Chamberland and Martelli are shown to be true for a subset of possible starting conditions.

Geometrical properties of the mean-median map

“…In the last section, we describe the results of exact computations on the system [0, x, 1], made possible by the theory just developed. We have established the strong terminating conjecture in specified neighbourhoods of 2791 rational numbers in the interval 1 2 , 2 3 , thereby extending the results of [2] by two orders of magnitude. This large data collection makes it clear that the domains surrounding the X-points where the limit function is regular do not account for the whole Lebesgue measure, suggesting the existence of a different, yet unknown, dynamical behaviour.…”

“…At such points, the sum (5) is finite. While local stabilisation -which holds in an open interval-is precisely what has been established near some rational points in the works mentioned earlier [11,2], global stabilisation is a much stronger property.…”

“…Then Y (1) (p n ) = Y (2) (q n ) = 1. In V n we have M n = Y (1) , Y (2) and M n−1 = Y (1) , and so by (4),…”

“…This multiset-enlarging rule is known as the mean-median map (mmm). This map and its iteration was introduced in [11], and subsequently studied in [3,2,6]. Such an iteration is meant to generate a data set whose mean and median coincide.…”

“…In a geometrical context, we shall also use the term X-point and the notation p = Y i ⊲⊳ Y j to mean the actual point on the plane rather than its abscissa. In the algorithm of [2], X-points appear as the endpoints of intervals where the combinatorics of the mmm orbits remains the same.…”