Spectral multiplicity for powers of weakly mixing automorphisms

V. V. Ryzhikov

Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 1065-1076

Résumé

We study the behaviour of the maximal spectral multiplicity $\mathfrak m(R^n)$ for the powers of a weakly mixing automorphism $R$. For some particular infinite sets $A$ we show that there exists a weakly mixing rank-one automorphism $R$ such that $\mathfrak m(R^n)=n$ and $\mathfrak m(R^{n+1})=1$ for all positive integers $n\in A$. Moreover, the cardinality $\operatorname{cardm}(R^n)$ of the set of spectral multiplicities for the power $R^n$ is shown to satisfy the conditions $\operatorname{cardm}(R^{n+1})=1$ and $\operatorname{cardm}(R^n)=2^{m(n)}$, $m(n)\to\infty$, $n\in A$. We also construct another weakly mixing automorphism $R$ with the following properties: all powers $R^{n}$ have homogeneous spectra and the set of limit points of the sequence $\{\mathfrak m(R^n)/n:n\in \mathbb N \}$ is infinite. Bibliography: 17 titles.

Détail

Keywords: weakly mixing transformation, homogeneous spectrum, maximal spectral multiplicity.

Bibliographie

[1] E. H. El Abdalaoui, “On the spectrum of the powers of Ornstein transformations”, Sankhya Ser. A, 62:3 (2000), 291–306 | MR | Zbl

[2] 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

[3] O. N. Ageev, “Dynamical systems with an even-multiplicity Lebesgue component in the spectrum”, Math. USSR-Sb., 64:2 (1989), 305–317 | DOI | MR | Zbl

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

[5] V. I. Oseledets, “The spectrum of ergodic automorphisms”, Soviet Math. Dokl., 7 (1966), 776–779 | MR | Zbl

[6] A. M. Stepin, “Sprectral properties of ergodic dynamical systems with locally compact time”, Soviet Math. Dokl., 7 (1966), 1007–1010 | MR | Zbl

[7] A. Katok, Combinatorial constructions in ergodic theory and dynamics, Univ. Lecture Ser., 30, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[8] A. Katok, J.-P. Thouvenot, “Spectral properties and combinatorial constructions in ergodic theory”, Handbook of dynamical systems, v. 1B, Elsevier, Amsterdam, 2006, 649–743 | MR | Zbl

[9] M. Lemańczyk, Spectral theory of dynamical systems, Encyclopedia of Complexity and System Science, Springer-Verlag, 2009

[10] A. I. Danilenko, A survey on spectral multiplicities of ergodic actions, arXiv: 1104.1961

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

[12] O. N. Ageev, “The spectral type of the rearrangements $T_{\alpha,\beta}$”, Sb. Math., 188:8 (1997), 1119–1152 | DOI | MR | Zbl

[13] O. N. Ageev, “Spectral rigidity of group actions: applications to the case $\operatorname{gr}(t,s;ts=st^2)$”, Proc. Amer. Math. Soc., 134:5 (2006), 1331–1338 | DOI | MR | Zbl

[14] A. I. Danilenko, “Weakly mixing rank-one transformations conjugate to their squares”, Studia Math., 187:1 (2008), 75–93 | DOI | MR | Zbl

[15] 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

[16] J.-P. Thouvenot, “Some properties and applications of joinings in ergodic theory”, Ergodic theory and its connections with harmonic analysis (Alexandria, 1993), London Math. Soc. Lecture Note Ser., 205, Cambridge Univ. Press, Cambridge, 1995, 207–235 | MR | Zbl

[17] S. V. Tikhonov, “Mixing transformations with homogeneous spectrum”, Sb. Math., 202:8 (2011), 1231–1252 | DOI | MR | Zbl