Sieving random iterative function systems

It is known that backward iterations of independent copies a contractive random Lipschitz function converge almost surely under mild assumptions. By sieving (or thinning) procedure based on adding to the functions time and space components, it possible construct scale invariant stochastic process. We study its distribution paths properties. In particular, we show càdlàg has finite total variation. also provide examples analyse various properties particular sieved iterative systems including perpetuities infinite Bernoulli convolutions, maximum, continued fractions.