  Linear fractional Galton-Watson branching processes in i.i.d.~random
environment are, on the quenched level, intimately connected to random
difference equations by the evolution of the random parameters of their linear
fractional marginals. On the other hand, any random difference equation defines
an autoregressive Markov chain (a random affine recursion) which can be
positive recurrent, null recurrent and transient and which, as the forward
iterations of an iterated function system, has an a.s.~convergent counterpart
in the positive recurrent case given by the corresponding backward iterations.
The present expository article aims to provide an explicit view at how these
aspects of random difference equations and their stationary limits, called
perpetuities, enter into the results and the analysis, especially in quenched
regime. Although most of the results presented here are known, we hope that the
offered perspective will be welcomed by some readers.
