M&m Sequences

On two conjectures for M&m sequences

“…The aim in this paper is to continue the exploration of mean-median sequences and the associated mean-median map. The class of mean-median sequences, the generation of which we shortly describe, was introduced by Schultz and Shiflett [3] and further analysed by Chamberland and Martelli [2], who introduced the mean-median map to aid their investigations, and also by Bonchev Bonchev [1]. The interest in the mean-median map lies in the fact that it is a very good example of a simple process yielding extremely complicated dynamics.…”

“…By applying an affine transformation, we can always reduce the case of arbitrary a < b < c to 0 < x < 1 (this is shown in [2] and [3], although in [3] they choose the different normalisation 0 < x < x + 1). So, from this point on, let us suppose that we start with 0 < x < 1 and consider the M&m sequence (x 4 , x 5 , .…”

Geometrical properties of the mean-median map

“…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.…”

“…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.…”

“…As noted in [11], the mmm commutes with all affine real transformations, so that the simplest non-trivial case -three distinct initial numbers-may be studied in full generality by considering the initial multiset [0, x, 1], with x ∈ (0, 1), which depends on a single variable. Here one finds already substantial difficulties, which are synthesised in the following conjectures [11,3].…”