Mixing transformations with homogeneous spectrum

S. V. Tikhonov

Sbornik. Mathematics, Tome 202 (2011) no. 8, pp. 1231-1252

Résumé

In the paper, the existence of mixing measure-preserving transformations with homogeneous spectrum of arbitrary multiplicity is proved. Bibliography: 20 titles.

Détail

Keywords: homogeneous spectrum, mixing transformations, actions by measure-preserving transformations.

Bibliographie

[1] O. N. Ageev, “On ergodic transformations with homogeneous spectrum”, J. Dynam. Control Systems, 5:1 (1999), 149–152 | DOI | MR | Zbl

[2] V. V. Ryzhikov, “Homogeneous spectrum, disjointness of convolutions, and mixing properties of dynamical systems”, Selected Russian Math., 1:1 (1999), 13–24

[3] O. Ageev, “The homogeneous spectrum problem in ergodic theory”, Invent. Math., 160:2 (2005), 417–446 | DOI | MR | Zbl

[4] A. I. Danilenko, “Explicit solution of Rokhlin's problem on homogeneous spectrum and applications”, Ergodic Theory Dynam. Systems, 26:5 (2006), 1467–1490 | DOI | MR | Zbl

[5] V. V. Ryzhikov, “Spectral multiplicities and asymptotic operator properties of actions with invariant measure”, Sb. Math., 200:12 (2009), 1833–1845 | DOI | MR | Zbl

[6] O. Ageev, “Mixing with staircase multiplicity functions”, Ergodic Theory Dynam. Systems, 28:6 (2008), 1687–1700 | DOI | MR | Zbl

[7] A. I. Danilenko, New spectral multiplicities for mixing transformations, arXiv: 0908.1640

[8] V. V. Ryzhikov, “Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions”, Sb. Math., 198:5 (2007), 733–754 | DOI | MR | Zbl

[9] S. V. Tikhonov, “A complete metric in the set of mixing transformations”, Sb. Math., 198:4 (2007), 575–596 | DOI | MR | Zbl

[10] V. A. Rokhlin, “Izbrannye voprosy metricheskoi teorii dinamicheskikh sistem”, UMN, 4:2 (1949), 57–128 | MR | Zbl

[11] P. R. Halmos, Lectures on ergodic theory, Mathematical Society of Japan, Tokyo, 1956 | MR | Zbl | Zbl

[12] A. I. Danilenko, A. V. Solomko, Ergodic Abelian actions with homogeneous spectrum, arXiv: 1001.2259

[13] M. Foreman, B. Weiss, “An anti-classification theorem for ergodic measure preserving transformations”, J. Eur. Math. Soc. (JEMS), 6:3 (2004), 277–292 | DOI | MR | Zbl

[14] D. V. Anosov, “O spektralnykh kratnostyakh v ergodicheskoi teorii”, Sovr. probl. matem., 3, MIAN, M., 2003, 3–85 | DOI | MR | Zbl

[15] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic theory, Grundlehren Math. Wiss., 245, Springer-Verlag, New York, 1982 | MR | MR | Zbl | Zbl

[16] V. V. Ryzhikov, “Mixing, rank, and minimal self-joining of actions with an invariant measure”, Russian Acad. Sci. Sb. Math., 75:2 (1993), 405–427 | DOI | MR | Zbl | Zbl

[17] V. V. Ryzhikov, “Pairwise $\varepsilon$-independence of the sets $T^iA$ for a mixing transformation $T$”, Funct. Anal. Appl., 43:2 (2009), 155–157 | DOI | MR

[18] E. Lehrer, B. Weiss, “An $\varepsilon$-free Rohlin lemma”, Ergodic Theory Dynam. Systems, 2:1 (1982), 45–48 | DOI | MR | Zbl

[19] S. A. Yuzvinskij, “Metric automorphisms with simple spectrum”, Soviet Math. Dokl., 8 (1967), 243–245 | MR | Zbl

[20] P. R. Halmos, “Approximation theories for measure preserving transformations”, Trans. Amer. Math. Soc., 55:1 (1944), 1–18 | DOI | MR | Zbl