BOUNDED-TO-1 FACTORS OF AN APERIODIC SHIFT OF FINITE-TYPE ARE 1-TO-1 ALMOST EVERYWHERE FACTORS ALSO

We show that if pi: SIGMA-G --> SIGMA-H is a bounded-to-1 factor map from an irreducible shift of finite type SIGMA-G with period p(G) to a shift of finite type SIGMA-H with period p(H), then there is a factor map pi-triple-over-dot: SIGMA-G --> SIGMA-H that is (p(G)/p(H)-to-1 almost everywhere. Moreover, if pi is right closing, then pi-triple-over-dot may be taken to be right closing also.