Dynamique du probl\`eme $3x+1$ sur la droite r\'eelle

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 $\mathbb{R}^{+}$. 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 $\mathbb{R}^{-}$, in connection with the 3x-1 problem. Finally, we analyze the mean behaviour of real iterations near infinity and present a method based on a heuristic argument by R. E. Crandall in the discrete case. It follows that the average growth rate of the iterates is close to $(2+\sqrt{3})/4$ under a condition of uniform distribution modulo 2.