Geodesic
A realization of the Pascal automorphism in the concatenation graph, and the function~$s_2(n)$
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 95-102

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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. 
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     author = {A. A. Lodkin and I. E. Manaev and A. R. Minabutdinov},
     title = {A realization of the {Pascal} automorphism in the concatenation graph, and the function~$s_2(n)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
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A. A. Lodkin; I. E. Manaev; A. R. Minabutdinov. A realization of the Pascal automorphism in the concatenation graph, and the function~$s_2(n)$. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXI, Tome 403 (2012), pp. 95-102. http://geodesic.mathdoc.fr/item/ZNSL_2012_403_a5/