Automorphisms of shift spaces and the Higman - Thompson groups : the one-sided case
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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).
Bleak , C P , Cameron , P J & Olukoya , F 2021 , ' Automorphisms of shift spaces and the Higman - Thompson groups : the one-sided case ' , Discrete Analysis , vol. 2021 , 15 . https://doi.org/10.19086/da.28243
Copyright © 2021 The Authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/).
DescriptionFunding: 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.
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