Solenoids and the origins of shift equivalence
by
Bob Williams
University of Texas and IAS

The term ``Williams conjecture'' is very kindly applied to a result that I thought I had proved in 1974. In fact, it is not true, as has just been shown by Kim-Roush. The story begins in 1967 when branched 1-manifolds, 1DBMs were introduced in a paper about 1-dimensional hyperbolic attractors, 1DHAs. In studying foliations defined on a subset of a manifold (sometimes called laminations) certain quotients are shown to be 1DBMs. We used this idea in 1968 to study Anosov diffeomorphisms-it is still used today to study pseudo-Anosov diffeomorphisms. Shift equivalence, ~ _s arose quite naturally as a way of relating a given (simple) map, say g, to its natural extension, or inverse limit, [^g]. It was shown in 1967 that topological conjugacy classes of diffeomorphisms on 1DHAs correspond to shift equivalence classes of immersions of 1DBMs induced on quotients of the type indicated above. The tool of ßtrong shift equivalence" \approx _s, was shown to be equivalent to ~ _s by a rather basic argument which can be used to show this equivalence in most "categories." In 1974 we were deluded in believing that we had found a way of showing this equivalence in the rather unnatural category of square matrices with nonnegative integral entries. Of course, for the symbolic dynamicist, this is quite a natural setting, if not a 'category'. \approx _s does indeed characterize topological conjugacy of subshifts of finite type.

Date received: February 24, 1998