Product Decay for Backward Trajectories, Staircase Systems, and Trees of Maps

We study product-decay conditions for backward trajectories in nonstationary fixed-point systems and revisit two statements from earlier work. Backward images of the terminal spaces are nested, so uniform convergence with respect to the terminal point holds if and only if their diameters tend to zero. Consequently, on bounded complete spaces, decay of the relevant products of Lipschitz constants is enough; summability of these products is not needed. We distinguish convergence in the first space from the existence of a compatible thread at every level and apply the same argument to staircase systems and prefix-dependent trees of maps. We also give error estimates, prove that the product condition is optimal for maps of the interval, and construct a staircase system with infinitely many expansive bonding maps. The nested-set and point-fibred arguments used here are classical; the contribution is their application to the precise summability assumptions in the cited nonstationary results, together with the necessary qualifications and sharp estimates.