Contractivity and ergodicity of the random map $x\mapsto|x-\theta|$

The long time behavior of the random map $x_n\mapsto x_{n+1}= |x_n-\theta_n|$ is studied under various assumptions on the distribution of the $\theta_n$. One of the interesting features of this random dynamical system is that for a single fixed deterministic $\theta$ the map is not a contraction, while the composition is almost surely a contraction if $\theta$ is chosen randomly with only mild assumptions on the distribution of the $\theta$'s. The system is useful as an explicit model where more abstract ideas can be explored concretely. We explore various measures of convergence rates, hyperbolically from randomness, and the structure of the random attractor.