Let (An,Bn) be a sequence of two-dimensional i.i.d. random variables with values in R+ X R. It induces the reflected affine stochastic recursion X_n^x. When An=1, this is the classical reflected random walk on the half-line, and without the absolute value, it becomes an iteration of random affine transformations, studied by various authors in the last decades. In this paper, we study existence and uniqueness of invariant measures, as well as recurrence of this process. An initial chapter displays the unpublished work of Martin Benda on local contractivity, which merit publicity and provide an important tool for studying stochastic iterations.