Image Contraction under Random Composition of a Fixed Function Family

Fix a collection of maps on a finite set and apply them repeatedly in a uniformly random order. The set of possible outputs, the image, can only shrink or stay the same at each step, never grow, but the rate at which it shrinks turns out to depend on a single spectral radius that one can read off the graph of reachable image sets. The key observation is that the successive image sets form a Markov chain on a finite graph whose vertices are the reachable image sets and whose edges record what each map does to each image set. The long-run behaviour is determined by the strongly connected components of this graph. When the process eventually collapses to a single point (synchronisation), we give an exact formula for the expected image size at every step and show that the approach to the limit is geometrically fast at a rate given by a spectral radius. This same rate controls the dynamics on pairs of points. We also derive a lower bound on the expected synchronisation time that is explicit in the maps and sharp in examples, an upper bound of order N log N when the maps are redrawn fresh at each step (in the fixed-family case reuse creates correlations that break the argument), and, when all non-singleton image sets happen to have the same size, the exact covariance structure of the process. Five problems remain open: we could not close Q2, the transfer of the O(N log N) bound from fresh maps to a fixed family.