ERGODIC PROPERTIES OF A DYNAMICAL SYSTEM ARISING FROM PERCOLATION THEORY

We consider the following dynamical system: take a d-dimensional real vector with positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by x(n) the result after n iterations. It is shown that for almost all x, lim(n-->infinity)x(n) not equal 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.