×

A realization of the Pascal automorphism in the concatenation graph and the sum-of-digits function \(s_2(n)\). (English. Russian original) Zbl 1338.37016

J. Math. Sci., New York 190, No. 3, 459-463 (2013); translation from Zap. Nauchn. Semin. POMI 403, 95-102 (2012).
Summary: A class of concatenation dynamical systems is introduced. Various automorphisms, including the Morse and Pascal authomorphisms, can be regarded as automorphisms of this class. In this realization, a natural number-theoretic interpretation of the problem whether the spectrum of an automorphisms 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 authomorphism.

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A30 Ergodic theorems, spectral theory, Markov operators
08A50 Word problems (aspects of algebraic structures)
05C90 Applications of graph theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. M. Vershik, ”Uniform algebraic approximation of shift and multiplication operators,” Dokl. Akad. Nauk SSSR, 259, No. 3, 526–529 (1981). · Zbl 0484.47005
[2] X. Méla and K. Petersen, ”Dynamical properties of the Pascal adic transformation,” Ergodic Theory Dynam. Systems, 25, 227–256 (2005). · Zbl 1069.37007
[3] P. Flajolet, P. Grabner, H. Prodinger, and R. Tichy, ”Mellin transforms and asymptotics: digital sums,” Theoret. Comput. Sci., 123, 291–314 (1994). · Zbl 0788.44004
[4] M. Drmota and M. Skalba, ”Sign-changes of the Thue-Morse fractal function and Dirichlet L-series,” Manuscripta Math., 86, 519–541 (1995). · Zbl 0828.11013
[5] M. Drmota and M. Skalba, ”Rarified sums of the Thue-Morse sequence,” Trans. Amer. math. Soc., 352, 609–642 (2000). · Zbl 0995.11017
[6] A. Vershik, F. Petrov, and P. Zatitskiy, ”Geometry and dynamics of admissible metrics in measure spaces,” arXiv:1205.1174v2. · Zbl 1261.37004
[7] S. Ferenczi, ”Measure-theoretic complexity of ergodic systems,” Israel J. Math., 100, 189–207 (1997). · Zbl 1095.28510
[8] A. A. Lodkin, I. E. Manaev, and A. R. Minabutdinov, ”Asymptotic behavior of the scaling entropy of the Pascal authomorphism,” Zap. Nauchn. Semin. POMI, 378, 58–72 (2010). · Zbl 1335.37002
[9] A. M. Verski, ”Orbit theory, locally finite permutations and Morse arithmetic,” in: Dynamical Numbers: Interpaly Between Dynamical Systems and Number Theory, Contemp. Math., 532, Amer. Math. Soc., Providence, Rhode Island (2010), pp. 115–136.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.