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arXiv · 1210.0042

Dynamics of a quasi-quadratic map

Assis Azevedo·Maria Carvalho·António Machiavelo
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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.

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Assis Azevedo, Maria Carvalho, António Machiavelo. 2012-09-28. Dynamics of a quasi-quadratic map. https://arxiv.org/abs/1210.0042

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