9 problems

For an initial set of size three, use affine invariance to reduce to the initial set $[0,x,1]$ with $x\in\left[\frac{1}{2},\frac{2}{3}\right]$. Let $m(x)\in\mathbb{R}$ be the limit…

For an initial set of size three, use affine invariance to reduce to the initial set $[0,x,1]$ with $x\in\left[\frac{1}{2},\frac{2}{3}\right]$. Let…

Let $[x_1,\ldots,x_{n_0}]$ be a finite non-empty real multiset, and let $(x_n)_{n=1}^{\infty}$ be its mean-median map orbit. Write $\mathcal{M}_n$ for the median of…

Let $[x_1,\ldots,x_{n_0}]$ be a finite non-empty real multiset, and let the mean-median map adjoin $x_{n+1}$ so that the arithmetic mean of the enlarged multiset equals the median…

Finite transit-time conjecture. For every initial set, the sequence of new numbers generated by iterating the mean-median map is eventually constant.

For every odd integer $t\geqslant 9$, let $N_t$ be the index … where … and let $m_t$ be the limit of the normal form orbit and $y_{N_t}$ its term at index $N_t$. The upper-bound co…

For every odd integer $t\geqslant 5$, let $(y_n)$ be the normal form orbit of order $t$. The stabilisation conjecture for normal form orbits. The orbit $(y_n)$ stabilises. Normal f…

For an initial condition $x\in[0,1]$, let $(x_n)$ be the orbit generated by the mean-median map from the initial set $[0,x,1]$. The strong terminating conjecture. For every…

Starting from the normalized initial values $0<x<1$, let $(x_4,x_5,\ldots)$ be the associated M&m sequence, and let $L(x)$ denote the minimum integer $k$ such that $x_n=x_k$ for al…