Dynamics of a quasi-quadratic map

Abstract

We consider the map X : Q --> Q given by X(x) = inverted right perpendicularxinverted left perpendicular, where inverted right perpendicular x inverted left perpendicular denotes the smallest integer greater than or equal to x, and study the problem of finding, for each rational, the smallest number of iterations by x that sends it into an integer. Given two natural numbers M and n, we prove that the set of numerators of the irreducible fractions that have denominator M and whose orbits by x reach an integer in exactly n iterations is a disjoint union of congruence classes modulo Mn+1. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide whether an orbit of a rational number bigger than one fails to hit an integer until a prescribed number of iterations have elapsed, and deduce that the probability that such an orbit enters Z is equal to 1.