Bounded ergodic constructions, disjointness, and weak limits of powers

Author:
V. V. Ryzhikov

Translated by:
E. Khukhro

Original publication:
Trudy Moskovskogo Matematicheskogo Obshchestva, tom **74** (2013), vypusk 1.

Journal:
Trans. Moscow Math. Soc. **2013**, 165-171

MSC (2010):
Primary 28D05; Secondary 58F11

Published electronically:
April 9, 2014

MathSciNet review:
3235793

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the disjointness property of powers of a totally ergodic bounded construction of rank 1 and some generalizations of this result. We look at applications to the problem when the Möbius function is independent of the sequence induced by a bounded construction.

**1.**Harry Furstenberg,*Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation*, Math. Systems Theory**1**(1967), 1–49. MR**0213508****2.**J. Bourgain, P. Sarnak, and T. Ziegler,*Disjointness of Moebius from horocycle flows*, From Fourier analysis and number theory to Radon transforms and geometry, Dev. Math., vol. 28, Springer, New York, 2013, pp. 67–83. MR**2986954**, 10.1007/978-1-4614-4075-8_5**3.**P. Sarnak,*Three lectures on the Mobius function randomness and dynamics*,

#publications.ias.edu/sarnak/#.**4.**J. Bourgain,*On the correlation of the Moebius function with random rank-one systems*, #arXiv:1112.1032#.**5.**V. V. Ryzhikov,*Minimal self-joinings, bounded constructions, and weak closure of ergodic actions*, #arXiv:1212.2602#.**6.**H. Abdalaoui, M. Lemanczyk, and T. De La Rue,*On spectral disjointness of powers for rank-one transformations and Moebius orthogonality*, #arXiv:1301.0134#.**7.**V. V. Ryzhikov,*Simple spectrum of the tensor product of powers of a mixing automorphism*, Trudy Mosk. Mat. Ob-va**73**(2012), no. 2, 229-239; English transl., Trans. Mosc. Math. Soc.**2012**(2012), 183-191.**8.**A. A. Prikhod'ko,*Littlewood polynomials and their appliucations to the spectral theory of dynamical systems*. Mat. Sb.**204**(2013), no. 6, 135-160; English transl., Sb. Math.**204**(2013).**9.**V. V. Ryzhikov,*Wreath products of tensor products, and a stochastic centralizer of dynamical systems*, Mat. Sb.**188**(1997), no. 2, 67–94 (Russian, with Russian summary); English transl., Sb. Math.**188**(1997), no. 2, 237–263. MR**1453260**, 10.1070/SM1997v188n02ABEH000202**10.**Oleg N. Ageev,*Spectral rigidity of group actions: applications to the case 𝑔𝑟⟨𝑡,𝑠;𝑡𝑠=𝑠𝑡²⟩*, Proc. Amer. Math. Soc.**134**(2006), no. 5, 1331–1338 (electronic). MR**2199176**, 10.1090/S0002-9939-05-08380-2**11.**Alexandre I. Danilenko,*Weakly mixing rank-one transformations conjugate to their squares*, Studia Math.**187**(2008), no. 1, 75–93. MR**2410884**, 10.4064/sm187-1-4**12.**V. V. Ryzhikov,*Spectral multiplicities of powers of a weakly mixing automorphism*, Mat. Sb.**203**(2012), no. 7, 149–160 (Russian, with Russian summary); English transl., Sb. Math.**203**(2012), no. 7-8, 1065–1076. MR**2986435**, 10.1070/SM2012v203n07ABEH004254

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Additional Information

**V. V. Ryzhikov**

Affiliation:
Moscow State University

Email:
vryzh@mail.ru

DOI:
https://doi.org/10.1090/S0077-1554-2014-00214-4

Keywords:
Ergodic power of a transformation,
construction of rank one,
disjointness of dynamical systems,
M\"obius function.

Published electronically:
April 9, 2014

Additional Notes:
This research was supported by the grant NSh-5998.2012.1.

Article copyright:
© Copyright 2014
V. V. Ryzhikov