Eigenvalues and Transduction of Morphic Sequences
We study finite state transduction of automatic and morphic sequences. Dekking  proved that morphic sequences are closed under transduction and in particular morphic images. We present a simple proof of this fact, and use the construction in the proof to show that non-erasing transductions preserve a condition called α-substitutivity. Roughly, a sequence is α-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue α. Our results culminate in the following fact: for multiplicatively independent real numbers α and β, if v is a α-substitutive sequence and w is an β-substitutive sequence, then v and w have no common non-erasing transducts except for the ultimately periodic sequences. We rely on Cobham’s theorem for substitutions, a recent result of Durand .
KeywordsIncidence Matrix Dominant Eigenvalue Input Alphabet Morphic Image Output Alphabet
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