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On the collection of spectral multiplicities \(\{2, 4,\dots,2^n\}\) for totally ergodic \(\mathbb Z^2\)-actions. (English. Russian original) Zbl 1370.37062
Math. Notes 96, No. 3, 360-368 (2014); translation from Mat. Zametki 96, No. 3, 383-392 (2014).
Summary: The paper is devoted to the realization of collections of spectral multiplicities for ergodic \(\mathbb Z^2\)-actions. Sufficient conditions ensuring the realizability of multiplicities of the form \(\{2,4,\dots,2^n\}\) are given.

37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37A30 Ergodic theorems, spectral theory, Markov operators
Full Text: DOI
[1] Anosov, D V, On spectral multiplicity of ergodic theories, 3-85, (2003), Moscow · Zbl 1071.37003
[2] Goodson, G R, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems, 5, 173-226, (1999) · Zbl 0987.37004
[3] Danilenko, A I, A survey on spectral multiplicities of ergodic actions, Ergodic Theory Dynam. Systems, 33, 81-117, (2013) · Zbl 1261.37009
[4] Ryzhikov, V V, Transformations having homogeneous spectra, J. Dynam. Control Systems, 5, 145-148, (1999) · Zbl 0954.37007
[5] Ageev, O N, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems, 5, 149-152, (1999) · Zbl 0943.37005
[6] Ryzhikov, V V, Spectral multiplicities and asymptotic operator properties of actions with invariantmeasure, Mat. Sb., 200, 107-120, (2009)
[7] Katok, A; Lemańczyk, M, Some new cases of realization spectral multiplicity function for ergodic transformations, Fund. Math., 206, 185-215, (2009) · Zbl 1187.37005
[8] Danilenko, A I; Lemańczyk, M, Spectral multiplicities for ergodic flows, Discrete Contin. Dyn. Syst., 33, 4271-4289, (2013) · Zbl 1306.37006
[9] Solomko, A V, New spectral multiplicities for ergodic actions, Studia Math., 208, 229-247, (2012) · Zbl 1254.37003
[10] Ryzhikov, V V, Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions, Mat. Sb., 198, 137-159, (2007) · Zbl 1161.37011
[11] Tikhonov, S V, Mixing transformations with homogeneous spectrum, Mat. Sb., 202, 139-160, (2011) · Zbl 1247.37008
[12] Stepin, A M, Spectral properties of generic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 50, 801-834, (1986)
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