Periodic seeded arrays and automorphisms of the shift

Abstract:

The automorphism group $\operatorname {Aut}({\Sigma _2})$ of the full $2$-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup $\mathcal {H}$ of $\operatorname {Aut}({\Sigma _2})$ ; $\mathcal {H}$ is generated by certain involutions ${g_n}$, and turns out to have a number of curious properties. For example, ${g_n}$ and ${g_k}$ commute unless $n$ and $k$ are consecutive integers, the order of ${g_{n + k}} \circ \cdots \circ {g_k}$ is independent of $k$, and $\mathcal {H}$ contains elements of all orders. The investigation is aided by the development of results about certain new types of arrays of $0$’s and $1$’s called periodic seeded arrays, as well as the use of Boyle and Krieger’s work on return numbers and periodic points.