Automorphisms of shift spaces and the Higman - Thompson groups: the one-sided case

Let 1 ≤ r < n be integers. We give a proof that the group Aut(X^ℕ_n,σ_n) of automorphisms of the one-sided shift on n letters embeds naturally as a subgroup ℋ_n of the outer automorphism group Out(G_n,r) of the Higman-Thompson group G_n,r. From this, we can represent the elements of Aut(X^ℕ_n,σ_n) by finite state non-initial transducers admitting a very strong synchronizing condition.

Let H ∈ ℋ_n and write |H| for the number of states of the minimal transducer representing H. We show that H can be written as a product of at most |H| torsion elements. This result strengthens a similar result of Boyle, Franks and Kitchens, where the decomposition involves more complex torsion elements and also does not support practical a priori estimates of the length of the resulting product.

We also explore the number of foldings of de Bruijn graphs and give acounting result for these for word length 2 and alphabet size n.

Finally, we offer new proofs of some known results about Aut(X^ℕ_n,σ_n).