Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences

Aedo Goñi, Ibai (2023). Forward Limit Sets of Semigroups of Substitutions and Arithmetic Progressions in Automatic Sequences. PhD thesis The Open University.



This thesis deals with symbolic sequences generated by semigroups of substitutions acting on finite alphabets.

First, we investigate the underlying structure of certain automatic sequences by studying the maximum length A(d) of the monochromatic arithmetic progressions of difference d appearing in these sequences. For example, for the Thue-Morse sequence and a class of generalised Thue-Morse sequences, we give exact values of A(d) or upper bounds on it, for certain differences d. For aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue-Morse and Rudin-Shapiro substitutions, respectively, we study the asymptotic growth rate of A(d). In particular, we prove that there exists a subsequence (d_n) of differences along which A(d_n) grows at least polynomially in d_n. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution considered.

Next, we introduce the forward limit set Λ of a semigroup S generated by a family of substitutions of a finite alphabet, which typically coincides with the set of all possible s-adic limits of that family. We provide several alternative characterisations of the forward limit set. For instance, we prove that Λ is the unique maximal closed and strongly S-invariant subset of the space of all infinite words, and we prove that it is the closure of the image under S of the set of all fixed points of S. It is usually difficult to compute a forward limit set explicitly; however, we show that, provided certain assumptions hold, Λ is uncountable, and we supply upper bounds on its size in terms of logarithmic Hausdorff dimension.

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