A few things are known, and many are unknown, on the automorphism group of the free MV-algebra over n - 1 generators. In this paper we show that this group appears as the stabilizer of 11 in the larger group of all automorphisms of the free cancellative hoop over n generators. Both groups have a dual action on the same space, namely the (n-1)-dimensional cube. The larger group has a richer dynamics, at the expense of loosing the two key features of the McNaughton homeomorphisms: preservation of denominators of rational points, and preservation of the Lebesgue measure. We present here some basic results, some examples, and some problems.
