Presentations of Schützenberger groups of minimal subshifts

Almeida, Jorge; Costa, Alfredo

doi:10.1007/s11856-012-0139-4

Cited by 20 publications

(26 citation statements)

References 28 publications

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“…We also show that for a recurrent set S and a strongly connected automaton A, the set of elements of the transition monoid M of minimal S-rank is included in a D-class of M called its S-minimal D-class (Proposition 3.2). This regular D-class is unique when S is minimal and it is related with the results of [1] and [2] on the regular J -classes of free profinite semigroups.…”

Section: Introductionsupporting

confidence: 52%

Codes and Automata in Minimal Sets

Perrin¹

2015

Lecture Notes in Computer Science

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Section: Introductionsupporting

confidence: 52%

Codes and Automata in Minimal Sets

Perrin¹

2015

Lecture Notes in Computer Science

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“…As applications of recognizability, let us quote estimates on the number of invariant measures, on the rank, characterization of spectral eigenvalues, applications to automorphism groups [DDMP16] and to the Schützenberger group of minimal substitutive shifts [AC13], and in the context of tiling spaces and related aperiodicity issues, see also [Sol98,HRS05,FS14,AR16].…”

Section: Introductionmentioning

confidence: 99%

Recognizability for sequences of morphisms

Berthé¹,

Steiner²,

Thuswaldner³

2018

Ergod. Th. Dynam. Sys.

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We investigate different notions of recognizability for a free monoid morphism σ : A * → B * . Full recognizability occurs when each (aperiodic) point in B Z admits at most one tiling with words σ(a), a ∈ A. This is stronger than the classical notion of recognizability of a substitution σ : A * → A * , where the tiling must be compatible with the language of the substitution. We show that if |A| = 2, or if σ's incidence matrix has rank |A|, or if σ is permutative, then σ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé (1992) and Bezuglyi, Kwiatkowski and Medynets (2009), by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an S-adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the S-adic shift it generates, and the measurable Bratteli-Vershik dynamical system that it defines.

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“…See Allouche and Shallit (1999) for a survey on this topic, including several further connections with other branches of Mathematics. The first author and other collaborators have previously studied the sequence t in the framework of symbolic dynamics and its connections with free profinite semigroups (see Almeida and Costa (2013) and Almeida et al (2020)). It was in fact an attempt to construct a profinite semigroup with certain properties that prompted this work, although no further references to profinite semigroups will be made in this paper.…”

Section: Introductionmentioning

confidence: 99%

Binary patterns in the Prouhet-Thue-Morse sequence

Almeida¹,

Klíma²

2021

Discrete Mathematics &Amp; Theoretical Computer Science

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We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.

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Presentations of Schützenberger groups of minimal subshifts

Cited by 20 publications

References 28 publications

Codes and Automata in Minimal Sets

Codes and Automata in Minimal Sets

Recognizability for sequences of morphisms

Binary patterns in the Prouhet-Thue-Morse sequence

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