Dynamics of a quasi-quadratic map

• Type: Article in International Scientific Journal
• Date: 2014

Authors

Assis Azevedo (Author) Other; Maria Carvalho (Author) FCUP; Antonio Machiavelo (Author) FCUP

Journal

Title: Journal of Difference Equations and Applications

Vol. 20 No. 1

Pages: 36-48

ISSN: 1023-6198

Publisher: Taylor & Francis

Indexing

ISI Web of Knowledge - 1 Citation

Scopus - 0 Citations

Scientific classification

FOS: Natural sciences > Mathematics

CORDIS: Physical sciences > Mathematics > Number theory

Other information

Authenticus ID: P-008-FKS

DOI: 10.1080/10236198.2013.805754

Abstract (EN): 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 13