Vershik, A. M. The Pascal automorphism has a continuous spectrum. (English. Russian original) Zbl 1271.37007 Funct. Anal. Appl. 45, No. 3, 173-186 (2011); translation from Funkts. Anal. Prilozh. 45, No. 3, 16-33 (2011). Summary: In this paper, we describe the Pascal automorphism and present a sketch of the proof that its spectrum is continuous on the orthogonal complement of the constants. Cited in 12 Documents MSC: 37A30 Ergodic theorems, spectral theory, Markov operators Keywords:Pascal automotphism; continuous spectrum; entropy PDFBibTeX XMLCite \textit{A. M. Vershik}, Funct. Anal. Appl. 45, No. 3, 173--186 (2011; Zbl 1271.37007); translation from Funkts. Anal. Prilozh. 45, No. 3, 16--33 (2011) Full Text: DOI References: [1] V. I. Arnold, Experimental Observation of Mathematical Facts [in Russian], MTsNMO, Moscow, 2006. [2] V. Arnold, ”Statistics of the symmetric group representations as a natural science question on asymptotic of Young diagrams,” in: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, Amer. Math. Soc. Transl. Ser. 2, vol. 217, 2006, 1–8. · Zbl 1171.20305 [3] A. M. Vershik, ”Uniform algebraic approximations of shift and multiplication operators,” Dokl. Akad. Nauk SSSR, 259: 3 (1981), 526–529; English transl.: Soviet Math. Dokl., 24 (1981), 97–100. · Zbl 0484.47005 [4] A. M. Vershik, ”A theorem on periodical Markov approximation in ergodic theory,” Zap. Nauch. Sem. LOMI, 115 (1982), 72–82; English transl.: J. Soviet Math., 28 (1985), 667–674. · Zbl 0505.47006 [5] A. Vershik, ”Dynamics of metrics in measure spaces and their asymptotic invariants,” Markov Processes Related Fields, 16:1 (2010), 169–185. · Zbl 1203.37020 [6] A. M. Vershik, ”Scaling entropy and automorphisms with purely point spectrum,” Algebra i Analiz, 23:1 (2011), 111–135. [7] A. Hajian, Y. Ito, S. Kakutani, ”Invariant measure and orbits of dissipative transformations,” Adv. in Math., 9:1 (1972), 52–65. · Zbl 0236.28010 [8] S. Kakutani, ”A problem of equidistribution on the unit interval [0, 1],” in: Lecture Notes in Math., vol. 541, Springer-Verlag, Berlin, 1976, 369–375. · Zbl 0363.60023 [9] A. Kushnirenko, ”On metric invariants of entropy type,” UspekhiMat. Nauk, 22:5(137) (1967), 57–65; English transl.: Russian Math. Surveys, 22:5 (1967), 53–61. · Zbl 0169.46101 [10] A. A. Lodkin, I. E. Manaev, and A. R. Minabutdinov, ”Asymptotic behavior of the scaling entropy of the Pascal adic transformation,” Zap. Nauchn. Sem. POMI, 378 (2010), 58–72; English transl.: J. Math. Sci., 174:1 (2011), 28–35. · Zbl 1335.37002 [11] A. Vershik, ”Orbit theory, locally finite permutations and Morse arithmetic,” in: Dynamical Numbers: Interplay between Dynamical Systems and Number Theory, Contemp. Math., vol. 532, 2010, 115–136. · Zbl 1217.37002 [12] A. M. Vershik and S. V. Kerov, ”Combinatorial theory and the K-functor,” in: Itogi Nauki i Tekhn. Sovrem. Probl. Nov. Dost., vol. 26, VINITI, Moscow, 1975, 3–56; English transl.: J. Soviet Math., 38 (1987), 1701–1733. [13] An. A. Muchnik, Yu. L. Pritykin, and A. L. Semenov, ”Sequences close to periodic,” Uspekhi Mat. Nauk, 64:5(389) (2009), 21–96; English transl.: Russian Math. Surveys, 64:5 (2009), 805–871. · Zbl 1208.03017 [14] K. Petersen and K. Schmidt, ”Symmetric Gibbs measures,” Trans. Amer. Math. Soc., 349:7 (1997), 2775–2811. · Zbl 0873.28008 [15] S. Ferenczi, ”Measure-theoretic complexity of ergodic systems,” Israel Math. J., 100 (1997), 180–207. · Zbl 1095.28510 [16] X. Mela, Dynamical properties of the Pascal adic and related systems. PhD Thesis, Univ. North Carolina at Chapel Hill, 2002. [17] É. Janvresse and T. de la Rue, ”The Pascal adic transformation is loosely Bernoulli,” Ann. Inst. H. Poincaré (B), Prob. Statist., 40:2 (2004), 133–139. · Zbl 1044.28012 [18] X. Mela and K. Petersen, ”Dynamical properties of the Pascal adic transformation,” Ergodic Theory Dynam. Systems, 25:1 (2005), 227–256. · Zbl 1069.37007 [19] É. Janvresse, T. de la Rue, and Y. Velenik, ”Self-similar corrections to the ergodic theorem for the Pascal-adic transformation,” Stoch. Dyn., 5:1 (2005), 1–25. · Zbl 1072.37006 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.