1) The numbers in the display will march off to infinity and the calculator will overflow;

2) The numbers in the display will enter a cycle of length n, where n is 2 or larger, but not larger than the number of internal states of the calculator; or

3) The numbers will converge to a fixed point, x*.

Depending on the calculator design, the number of states might be the same as the number of distinct values that can be displayed. (Some calculators keep one or two extra digits after the decimal which are never displayed.)

It is not hard to discover functions which have fixed points that can be found by the calculator method. Such functions are called contraction mappings, and are well studied in the mathematical literature. Sir Isaac Newton showed how to discover the roots (zeroes) of a function, f(x), by constructing a new function, F(x) = x - f(x)/f'(x).

The roots of a function, f(x), are the values of x for which f(x) = 0.

(Here, f'(x) denotes the first derivative of f(x).) If the original function, f(x), is "reasonably well-behaved," F(x) will be a contraction mapping with fixed points at the roots of f(x). Newton's Method is perhaps the most famous application of the Fixed Point Theorem, and it is very easy to write an algorithm (computer program) to implement it. (As an exercise, it is instructive to find simple functions, f(x), for which Newton's method cycles without converging to the root, or diverges to infinity.)

The process of random perturbations is extensively studied in the literature. Norbert Weiner is best known for his study of random walks. To my mind, it is not a coincidence that Weiner was also a pioneer in the field of cybernetics (the merger of psychology, feedback control theory, and automatic computation). Weiner's last book was entitled, God and Golem. ("Golem" is a Yiddish word for an anthropomorphic robot.)

For the benefit of Google search engines, other keywords and phrases connected to this essay include: Teleonomy, Extropy and Entropy Gradient Reversal.