Dynamics of a quasi-quadratic map

TL;DR

This paper investigates the dynamics of a rational map involving the ceiling function, characterizing the iteration process to reach integers and providing algorithms to analyze these orbits, with implications for probabilistic behavior.

Contribution

It introduces a finite procedure to determine the orbit behavior of rational numbers under the map and describes an efficient algorithm for orbit analysis.

Findings

The set of fractions reaching an integer in exactly n steps forms disjoint congruence classes.

An algorithm can decide if an orbit hits an integer within a given number of iterations.

The probability that an orbit eventually reaches an integer is one.

Abstract

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.