Coboundaries and eigenvalues of finitary <inline-formula><tex-math id="M1">$ S $</tex-math></inline-formula>-adic systems

An $ S $-adic system is a symbolic dynamical system generated by iterating an infinite sequence of substitutions or morphisms, called a directive sequence. A finitary $ S $-adic dynamical system is one where the directive sequence consists of morphisms selected from a finite set. We study eigenvalues and coboundaries for finitary recognizable $ S $-adic dynamical systems, i.e., those where points can be uniquely desubstituted using the given sequence of morphisms. To do this we identify the notions of straightness and essential words, and use them to define a coboundary, inspired by Host's formalism, which allows us to express necessary and sufficient conditions that a complex number must satisfy in order to be a continuous or measurable eigenvalue. We then apply our results to finitary directive sequences of substitutions of constant length, and show how to create constant-length $ S $-adic shifts with non-trivial coboundaries. We show that in this case all continuous eigenvalues are rational and we give a complete description of the rationals that can be an eigenvalue, indicating how this leads to a Cobham-style result for these systems.