A realization of the Pascal automorphism in the concatenation graph, and the function $s_2(n)$

A class of concatenation dynamical systems is introduced. Various automorphisms, including the Morse and 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 nondiscreteness of the spectrum of the Morse automorphism and to give a new formulation of the discreteness problem in the case of the Pascal automorphism.