Dynamics of a quasi-quadratic map

Abstract

We consider the map $\cchi:\Q\to\Q$ given by $ \cchi(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 by $\cchi$ 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 $\cchi$ reach an integer in exactly $n$ iterations is a disjoint union of congruence classes modulo $M^{n+1}$. Moreover, we establish a finite procedure to determine them. We also describe an efficient algorithm to decide if 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 one.