Coboundaries and eigenvalues of morphic subshifts

We define a morphic subshift as a subshift generated by the image of a substitution subshift by another substitution. In other words, it is the subshift associated with a ultimately periodic directive sequence. We present an efficient algorithm for computing eigenvalues of morphic subshifts using coboundaries. We show that continuous eigenvalues of S-adic subshifts for primitive directive sequences are always associated with some coboundary. For morphic subshifts, we provide a characterization of the dimension of the Q-vector space generated by eigenvalues. Additionally, if the substitutions are unimodular and without coboundaries, we give a necessary and sufficient condition for weak mixing of the subshift.