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Spectrum and Factors of Substitution Dynamical Systems Open Access

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This dissertation studies topological factors substitutions, especially substitutions with constant length p. If p is prime we show that any factor is topologically conjugate to a substitution of length ps. This extends a previous result on the relationship between substitutions of constant length and automatic sequences, and it also complements a result of Mentzen about metric factors of constant length substitutions. We also prove a topological version of a result by Host and Parreau; we show that factors of constant length substitutions onto substitutions can usually be reduced to 2-block codes. Our main result is a counterexample to a conjecture of Baake that any substitution has a subshift factor that is metrically isomorphic to its maximal equicontinuous factor. If true, the conjecture would imply that there is always a continuous complex function depending on finitely-many coordinates such that the corresponding diffraction measure contains the entire discrete spectrum. Unfortunately this turns-out to be false, as the example shows by using the structure of bijective substitutions and the previous results to limit the factors which must be considered. Finally, we show that a partial result to the contrary holds for non-constant length substitutions having irrational eigenvalues, and we examine visual approximations of the diffraction spectra of our examples and other well-known substitutions.

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