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

Funding: The authors are all grateful for support from EPSRC research grant EP/R032866/1; the third author also gratefully acknowledges support from Leverhulme Trust Research Project Grant RPG-2017-159.Let 1 ≤ r < n be integers. We give a proof that the group Aut(Xℕn,σn) of automorphisms of the one-sided shift on n letters embeds naturally as a subgroup ℋn of the outer automorphism group Out(Gn,r) of the Higman-Thompson group Gn,r. From this, we can represent the elements of Aut(Xℕn,σn) by finite state non-initial transducers admitting a very strong synchronizing condition. Let H ∈ ℋn 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(Xℕn,σn).Publisher PDFPeer reviewe

Symbolic dynamics and the stable algebra of matrices

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as well as various specific rings. This algebra is of independent interest and can be followed with little attention to the symbolic dynamics. We include strong connectionsto algebraic K-theory and the inverse spectral problem for nonnegative matrices. We also review key features of the automorphism group of a shift of finite type, and the work of Kim, Roush and Wagoner giving counterexamples to Williams' Shift Equivalence Conjecture.Comment: 121 pages. Main changes from version 1: Author and subject indices were added. Various citations were added, with commentary. Bibliography items are now listed with internal references (i.e., pages of the paper on which they are cited