Recursivity and eigen-behaviour

Recursive processes use the result of an operation for the same operation (von Foerster, 1984, illustration by von Foerster).

If the result of a first operation

x1 = Op (xo)

is processed again with the same operator

x2 = Op (x1)

it leads to the formula

lim Op(n) = Opo

The result of the limes operation might be endless, bi- or multistable. It also might lead to a definite stable value which von Foerster called eigen-behaviour. The eigen-behaviour of a recursive operation does not depend on the operand but on the operator.

Von Foerster described it by extracting the square root. Irrespective of the starting number (n>0), the result will always be 1.