A characterization of the minimal invariant sets of Alspach’s mapping

In 1981, Dale Alspach modified the baker’s transform to produce the first example of a nonexpansive mapping TT on a weakly compact convex subset CC of a Banach space that is fixed point free. By Zorn’s lemma, there exist minimal weakly compact, convex subsets of CC which are invariant under TT and are fixed point free.In this paper we produce an explicit formula for the nnth power of TT, TnTn, and prove that the sequence (Tnf)n∈N(Tnf)n∈N converges weakly to ‖f‖1χ[0,1], for all f∈Cf∈C. From this we derive a characterization of the minimal invariant sets of TT.