# zbMATH — the first resource for mathematics

Simple spectrum of the tensor product of powers of a mixing automorphism. (English. Russian original) Zbl 1275.37008
Trans. Mosc. Math. Soc. 2012, 183-191 (2012); translation from Tr. Mosk. Mat. O.-va 73, No. 2, 229-239 (2012).
The main result of the paper is the construction of a mixing automorphism $$T$$ such that the tensor product $$T\otimes T^2\otimes T^3\otimes\cdots$$ has simple spectrum. Here, by “mixing” it is meant the convergence $$T^n\to P$$, where $$P$$ is the orthoprojection onto the space of constants in $$L_2(X,\mu)$$. To prove the main result, the author provides constructions of rank-1 transformations with mixing property. Further, the simplicity of the spectrum of tensor product of powers of special nonmixing transformations is proved. Then, preserving the spectral property, the required assertion is proved.

##### MSC:
 37A30 Ergodic theorems, spectral theory, Markov operators 28D05 Measure-preserving transformations 47A35 Ergodic theory of linear operators
Full Text:
##### References:
 [1] Terrence M. Adams, Smorodinsky’s conjecture on rank-one mixing, Proc. Amer. Math. Soc. 126 (1998), no. 3, 739 – 744. · Zbl 0906.28006 [2] Oleg Ageev, The homogeneous spectrum problem in ergodic theory, Invent. Math. 160 (2005), no. 2, 417 – 446. · Zbl 1064.37003 · doi:10.1007/s00222-004-0422-z · doi.org [3] О спектрал$$^{\приме}$$ных кратностях в ѐргодической теории, Современные Проблемы Математики [Цуррент Проблемс ин Матхематицс], вол. 3, Российская Академия Наук, Математический Институт им. В. А. Стеклова, Мосцощ, 2003 (Руссиан, щитх Руссиан суммары). Аваилабле елецтроницаллы ат хттп://щщщ.ми.рас.ру/спм/пдф/003.пдф. [4] Darren Creutz and Cesar E. Silva, Mixing on a class of rank-one transformations, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 407 – 440. · Zbl 1066.37003 · doi:10.1017/S0143385703000464 · doi.org [5] A. I. Danilenko, A survey on spectral multiplicities of ergodic actions, Preprint, #arXiv:1104.1961#. [6] Andrés del Junco and Mariusz Lemańczyk, Generic spectral properties of measure-preserving maps and applications, Proc. Amer. Math. Soc. 115 (1992), no. 3, 725 – 736. · Zbl 0762.28015 [7] Donald S. Ornstein, On the root problem in ergodic theory, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 347 – 356. [8] A. A. Prikhod$$^{\prime}$$ko, Stochastic constructions of flows of rank 1, Mat. Sb. 192 (2001), no. 12, 61 – 92 (Russian, with Russian summary); English transl., Sb. Math. 192 (2001), no. 11-12, 1799 – 1828. · Zbl 1017.37002 · doi:10.1070/SM2001v192n12ABEH000616 · doi.org [9] V. V. Ryzhikov, Weak limits of powers, the simple spectrum of symmetric products, and mixing constructions of rank 1, Mat. Sb. 198 (2007), no. 5, 137 – 159 (Russian, with Russian summary); English transl., Sb. Math. 198 (2007), no. 5-6, 733 – 754. · Zbl 1161.37011 · doi:10.1070/SM2007v198n05ABEH003857 · doi.org [10] V. V. Ryzhikov, On mixing constructions with algebraic spacers, Preprint, #arXiv:1108.1508#. [11] A. M. Stëpin, Spectral properties of generic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 801 – 834, 879 (Russian). [12] S. V. Tikhonov, Mixing transformations with homogeneous spectrum, Mat. Sb. 202 (2011), no. 8, 139 – 160 (Russian, with Russian summary); English transl., Sb. Math. 202 (2011), no. 7-8, 1231 – 1252. · Zbl 1247.37008 · doi:10.1070/SM2011v202n08ABEH004185 · doi.org
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.