2022
DOI: 10.1007/s00454-022-00387-8
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Admissible Reversing and Extended Symmetries for Bijective Substitutions

Abstract: In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to compute them. Moreover, for any finite group H and any subgroup P of the d-dimensional hyperoctahedral group, we construct a bijective … Show more

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Cited by 4 publications
(2 citation statements)
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“…where the first equality holds by Fact 3, and the last equality holds because g |G| = id, for every group element g ∈ G. Since ̺ |G| has L columns equal to the identity substitution distributed in arithmetic progression of difference d, any fixed point of ̺ has a monochromatic arithmetic progression of difference d and length at least L. This completes the proof for k = 1. This proof extends to every positive integer k because, since ̺ has a column which is equal to the identity substitution (the leftmost column), the group generated by the columns of ̺ is equal to the group generated by the columns of ̺ k [9]. This means for a fixed k 1, one can take ̺ k|G| and construct d (k) , this time with 0 m L k − 1.…”
Section: Bijective Automatamentioning
confidence: 93%
See 1 more Smart Citation
“…where the first equality holds by Fact 3, and the last equality holds because g |G| = id, for every group element g ∈ G. Since ̺ |G| has L columns equal to the identity substitution distributed in arithmetic progression of difference d, any fixed point of ̺ has a monochromatic arithmetic progression of difference d and length at least L. This completes the proof for k = 1. This proof extends to every positive integer k because, since ̺ has a column which is equal to the identity substitution (the leftmost column), the group generated by the columns of ̺ is equal to the group generated by the columns of ̺ k [9]. This means for a fixed k 1, one can take ̺ k|G| and construct d (k) , this time with 0 m L k − 1.…”
Section: Bijective Automatamentioning
confidence: 93%
“…Here, we use a criterion for aperiodicity provided in [26,Prop. 4.1], which states that a sufficient condition for aperiodicity for primitive and bijective substitutions is the existence of two lengthtwo legal words which share either the same starting letter or the same ending letter; see also [9,Prop. 2.5].…”
mentioning
confidence: 99%