Dynamique du problème 3x + 1 sur la droite réelle

Abstract

The 3x + 1 problem is a difficult conjecture dealing with quite a simple algorithm on the positive integers. A possible approach is to go beyond the discrete nature of the problem, following M. Chamberland who used an analytic extension to the half-line R + . We complete his results on the dynamic of the critical points and obtain a new formulation the 3x + 1 problem. We clarify the links with the question of the existence of wandering intervals. Then, we extend the study of the dynamic to the half-line R − , in connection with the 3x − 1 problem. Finally, we analyze the mean behaviour of real iterations near ±∞. It follows that the average growth rate of the iterates is close to (2 + √ 3)/4 under a condition of uniform distribution modulo 2.