Continuous Spectra For Substitution-Based Sequences

Chan, Lax (2018). Continuous Spectra For Substitution-Based Sequences. PhD thesis The Open University.

DOI: https://doi.org/10.21954/ou.ro.0000d19d

Abstract

This thesis is chiefly concerned with the continuous spectra of substitution-based sequences. First, motivated by a question of Lafrance, Yee and Rampersad [34], we establish a connection between the ‘root-N’ property and the corresponding sequences that satisfy it having absolutely continuous spectrum. Then we use the recent advances in Bartlett [10, 11] to show that the Rudin–Shapiro-like sequence has singular continuous spectrum, hence does not satisfy the root-N property. This gives a negative answer to the question raised by the authors in [34].

Secondly, we use the connection we establish between the root-N property and absolute continuity to create more substitution-based sequences that have absolutely continuous/Lebesgue spectrum. This is done by modifying Rudin’s original construction [44]. We show that the binary sequences (±1 sequences) from our modification also satisfy the root-N property and they are mutually locally derivable to the corresponding substitution sequences. This shows that the spectral properties of the substitution-based sequences are inherited from their binary counterpart.

Finally, we generalise our construction using Fourier matrices. This leads to extending Rudin’s construction to sequences with complex coefficients. This approach allows us to generate substitution sequences of any constant length greater than or equal to two. We show explicitly in the length 3 and 4 cases that these systems exhibit Lebesgue spectrum, employing Bartlett’s algorithm from Chapter 3 and mutual local derivability.

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