Eigenvalues and Transduction of Morphic Sequences: Extended Version

TL;DR

This paper investigates how morphic sequences behave under finite state transduction, proving that certain properties are preserved and establishing conditions under which sequences share transducts, with implications for automatic and morphic sequences.

Contribution

The paper provides a new, simplified proof that morphic sequences are closed under transduction and extends this to show preservation of alpha-substitutivity under non-erasing transductions.

Findings

Morphic sequences are closed under transduction.

Non-erasing transductions preserve alpha-substitutivity.

Sequences with multiplicatively independent eigenvalues have no common non-erasing transducts except periodic ones.

Abstract

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 alpha-substitutivity. Roughly, a sequence is alpha-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue alpha. Our results culminate in the following fact: for multiplicatively independent real numbers alpha and beta, if v is an alpha-substitutive sequence and w is a beta-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.