2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140638In this paper, we study a class of quasi-invariant measures on paths generated by discrete dynamical systems. Our main result characterizes the subfamily of these measures which admit a certain desintegration. This is a desintegration with respect to a random walk Markov process which is indexed by the starting point of the paths. Our applications include wavelet constructions on Julia sets of rational maps on the Riemann sphere.new version minor additionsClassical Analysis and ODEsDynamical Systems42C40, 42A16, 42A65, 43A65, 46G15, 47D07, 60G18text