  We consider a model for which every site of $\mathbb{N}$ is assigned a
fitness in $[0,1]$. At every discrete time all the sites are updated and each
site samples a uniform on $[0,1]$, independently of everything else. At every
discrete time and independently of the past the environment is good with
probability $p$ or bad with probability $1-p$. The fitness of each site is then
updated to the maximum or the minimum between its present fitness and the
sampled uniform, according to whether the environment is good or bad. Assuming
the initial fitness distribution is exchangeable over the site indexing, the
empirical fitness distribution is a probability-valued Markov process. We show
that this Markov process converges to an explicitly-identified stationary
distribution exhibiting a self-similar structure.
