# American Institute of Mathematical Sciences

September  2013, 33(9): 4271-4289. doi: 10.3934/dcds.2013.33.4271

## Spectral multiplicities for ergodic flows

 1 Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov, 61164, Ukraine 2 Faculty of Mathematics and Computer Science, N. Copernicus University, ul. Chopina 12/18, 87-100 Toruń

Received  August 2010 Revised  February 2011 Published  March 2013

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.
Citation: Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
##### References:
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##### References:
 [1] O. N. Ageev, On ergodic transformations with homogeneous spectrum,, J. Dynam. Control Systems 5 (1999), 5 (1999), 149. Google Scholar [2] A. I. Danilenko, $(C,F)$-actions in ergodic theory,, Progr. Math. 265 (2008), 265 (2008), 325. Google Scholar [3] A. I. Danilenko, Explicit solution of Rokhlin's problem on homogeneous spectrum and applications,, Ergod. Th. & Dyn. Syst. 26 (2006), 26 (2006), 1467. Google Scholar [4] A. I. Danilenko, On new spectral multiplicities for ergodic maps,, Studia Math. 197 (2010), 197 (2010), 57. Google Scholar [5] A. I. Danilenko and S. V. Solomko, Ergodic Abelian actions with homogeneous spectrum,, Contemp. Math. 532 (2010), 532 (2010), 137. Google Scholar [6] I. Filipowicz, Product $Z^d$-actions on a Lebesgue space and their applications,, Studia Math. 122 (1997), 122 (1997), 289. Google Scholar [7] K. Frączek, Cyclic space isomorphism of unitary operators,, Studia Math. 124 (1997), 124 (1997), 259. Google Scholar [8] G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems,, J. Dynam. Control Systems 5 (1999), 5 (1999), 173. Google Scholar [9] E. Hewitt and K. A. Ross, "Abstract Harmonic Analysis'',, Vol. I, (1963). Google Scholar [10] A. Katok and M. Lemańczyk, Some new cases of realization of spectral multiplicity function for ergodic transformations,, Fund. Math. 206 (2009), 206 (2009), 185. Google Scholar [11] A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory,, in, (2006), 649. Google Scholar [12] R. A. Konev and V. V. Ryzhikov, On spectral multiplicities ${2,\ldots,2^n}$ for totally ergodic $\mathbbZ^2$-actions,, Preprint, (). Google Scholar [13] J. Kwiatkowski jr and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II,, Studia Math., 116 (1995), 207. Google Scholar [14] A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs,, Erg. Th. & Dyn. Syst., 7 (1987), 531. Google Scholar [15] M. Lemańczyk, Spectral theory of dynamical systems,, in, (2009), 8554. Google Scholar [16] M. Lemańczyk and F. Parreau, Special flows over irrational rotations with the simple convolutions property,, preprint., (). Google Scholar [17] G. Mackey, Induced representations of locally compact groups. I,, Ann. Math. 55 (1952), 55 (1952), 101. Google Scholar [18] V. V. Ryzhikov, Transformations having homogeneous spectra,, J. Dynam. Control Systems, 5 (1999), 145. Google Scholar [19] V. V. Ryzhikov, Spectral multiplicities and asymptotic operator properties of actions with an invariant measure,, Mat. Sb., 200 (2009), 107. Google Scholar [20] R. Zimmer, Induced and amenable ergodic actions of Lie groups,, Ann. Sci. Ecole Norm. Sup., 11 (1978), 407. Google Scholar
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