Abstract: Consider a proper metric space X and a sequence (Fn) of i.i.d.

random continuous mappings X → X. It induces the stochastic dynamical system (SDS)

Xn = Fn ◦ · · · ◦ F1(x) starting at x ∈ X.
We study existence and uniqueness of invariant measures, under some assumptions of

contraction on the Fn, as well as recurrence and ergodicity of this process.
We will consider two main examples: the case where the Fn are affine maps of the real line

and the case where Xn is the reflected random walk on the positive real line.