Approximate squaring and other iterations

This talk studies the map F(x) = x ceiling(x). For each rational r=p/q, with r >1, is it true that some iterate of r under F is an integer? The answer, concerning the exceptions, is: ``No, never? Well, hardly ever.''

Similar questions are considered for the ``approximate multiplication'' map F_r(x) = r ceiling(x), where r is a fixed rational number. Starting from an initial value x_o, is it true that some iterate of F_r(x) is an integer? The talk will explain analogies with the notorious 3x+1 problem. This is joint work with Neil Sloane (AT&T Labs); see math.NT/0309389.