Abstract

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 This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian.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.

Highlights

  • Let μ be the endomorphism of the free semigroup {a, b}+ defined by μ(a) = ab and μ(b) = ba

  • The sequence (μn(a))n determines a sequence of letters, or infinite word, whose prefix of length 2n is μn(a); we say that the infinite word t obtained is generated by μ

  • It was first considered by Prouhet (1851) in connection with a problem in number theory, five decades later by Thue (1906, 1912) to exhibit infinite words avoiding cubes and squares, and another two decades later by Morse (1921) as a discretized description of non-periodic recurrent geodesics in surfaces of negative curvature

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Summary

Introduction

The sequence (μn(a))n determines a sequence of letters, or infinite word, whose prefix of length 2n is μn(a); we say that the infinite word t obtained is generated by μ It is called the Prouhet-Thue-Morse sequence and it has been the object of extensive studies and applications. This paper concerns the study of binary patterns of t, that is, finite or infinite words w over the alphabet {a, b} for which there exists an endomorphism φ of the semigroup {a, b}+ (naturally extended to infinite words) such that the word φ(w) can be found as a block of consecutive letters of t (which we call a segment of t).

Segments of t
Finite binary patterns
Typical finite binary patterns
Final remarks and problems
Full Text
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