A class of concatenation dynamical systems is introduced. Various automorphisms, including
the Morse and the Pascal automorphisms, can be regarded as automorphisms of this class. In this
realization, a natural number-theoretic interpretation of the problem whether the spectrum of an automorphism is discrete arises. In particular, the known character of the asymptotic behavior of
the function $s_2(n)$ allows one to immediately see the non-discreteness of the spectrum of the
Morse automorphism and to give a new formulation of the discreteness problem in the case of the
Pascal automorphism.