Title: Subtractive algorithms

Abstract: 

Let $d\geq 3$, and consider the set $\Omega_d$ of $d$-tupels $\bar{x}=(x_1, x_2,\dots , x_d)$, where the $x_i$ are positive real numbers, and where $x_1 \leq x_2\leq \cdot \leq x_d$. Consider the map $T_d:\Omega_d\to\Omega_d$, defined by 
$$ T_d(\bar{x}) = (y_1, y_2,\dots , y_d), $$ where the $y_i$ are positive real numbers, $y_1 \leq y_2\leq \cdot \leq y_d$ and where $\{ y_1,y_2,\dots , y_d\} = \{ x_1, x_2-x_1, \dots  ,x_d-x_1\}$. In words: subtract $x_1$ from all other coordinates, and re-order the result in ascending order.

Setting $\bar{x}_0=\bar{x}$ and $\bar{x}_n=T^n(\bar{x}_0)$ for $n\geq 1$, we obviously have that whenever the coordinates of $\bar{x}_0$ are dependent over the integers, after finitely many steps $n$ the first coordinate of $\bar{x}_n$ will equal $0$. But what happens if the coordinates of $\bar{x}_0$ are independent over the integers?

In this talk we will consider this and related questions on multi-dimensional continued fraction algorithms. 

Location: KdVI meeting room, Science Park 105-107, F3.20