All powers of a function can have non-homeomorphic inverse limits

In this paper, we construct an upper semicontinuous function f:[0,1]→2[0,1]f:[0,1]→2[0,1] for which the following holds true: lim←{[0,1],fi} and lim←{[0,1],fj} are homeomorphic if and only if i=ji=j. This solves a problem stated by W.T. Ingram.