Dynamics of a quasi-quadratic map

We consider the map X defined on the rational numbers given by x --> x * ceil(x), where ceil(x) denotes the smallest integer greater than or equal to x, and study the problem of finding, for each rational, the smallest number of iterations of X that eventually sends it into an integer. Given two natural numbers M and n, we prove that the set of irreducible fractions with denominator M whose orbits by X reach an integer in exactly n iterations is a disjoint union of congruence classes modulo M^n, establishing along the way a finite procedure to ascertain them. We also describe an efficient algorithm to decide if an orbit fails to hit an integer until a prescribed number of iterations, and deduce that the probability that an orbit enters the set of integers is equal to one.