Spectral multiplicities for ergodic flows

Alexandre I. Danilenko; Mariusz Lemańczyk

doi:10.3934/dcds.2013.33.4271

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

Spectral multiplicities for ergodic flows

Alexandre I. Danilenko ^1, and Mariusz Lemańczyk ^2,

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

Abstract
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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.

Keywords: Ergodic flow, spectral multiplicities..

Mathematics Subject Classification: Primary: 37A10, 37A3.

Citation: Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271

References:

[1]	O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149-152.
[2]	A. I. Danilenko, $(C,F)$-actions in ergodic theory, Progr. Math. 265 (2008), 325-351.
[3]	A. I. Danilenko, Explicit solution of Rokhlin's problem on homogeneous spectrum and applications, Ergod. Th. & Dyn. Syst. 26 (2006), 1467-1490.
[4]	A. I. Danilenko, On new spectral multiplicities for ergodic maps, Studia Math. 197 (2010), 57-68.
[5]	A. I. Danilenko and S. V. Solomko, Ergodic Abelian actions with homogeneous spectrum, Contemp. Math. 532 (2010), 137-148.
[6]	I. Filipowicz, Product $Z^d$-actions on a Lebesgue space and their applications, Studia Math. 122 (1997), 289-298.
[7]	K. Frączek, Cyclic space isomorphism of unitary operators, Studia Math. 124 (1997), 259-267.
[8]	G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173-226.
[9]	E. Hewitt and K. A. Ross, "Abstract Harmonic Analysis'', Vol. I, Springer-Verlag, Berlin-Göt-tingen-Heidelberg, 1963.
[10]	A. Katok and M. Lemańczyk, Some new cases of realization of spectral multiplicity function for ergodic transformations, Fund. Math. 206 (2009), 185-215.
[11]	A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems'', vol. 1B, Elsevier B. V., 2006, 649-743.
[12]	R. A. Konev and V. V. Ryzhikov, On spectral multiplicities ${2,\ldots,2^n}$ for totally ergodic $\mathbbZ^2$-actions,, Preprint, ().
[13]	J. Kwiatkowski jr and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, Studia Math., 116 (1995), 207-215
[14]	A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Erg. Th. & Dyn. Syst., 7 (1987), 531-557
[15]	M. Lemańczyk, Spectral theory of dynamical systems, in "Encyclopedia of Complexity and Systems Science'' Springer-Verlag, 2009, 8554-8575.
[16]	M. Lemańczyk and F. Parreau, Special flows over irrational rotations with the simple convolutions property,, preprint., ().
[17]	G. Mackey, Induced representations of locally compact groups. I, Ann. Math. 55 (1952), 101-139.
[18]	V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems, 5 (1999), 145-148.
[19]	V. V. Ryzhikov, Spectral multiplicities and asymptotic operator properties of actions with an invariant measure, Mat. Sb., 200 (2009), 107-120.
[20]	R. Zimmer, Induced and amenable ergodic actions of Lie groups, Ann. Sci. Ecole Norm. Sup., 11 (1978), 407-428.

show all references

References:

[1]	O. N. Ageev, On ergodic transformations with homogeneous spectrum, J. Dynam. Control Systems 5 (1999), 149-152.
[2]	A. I. Danilenko, $(C,F)$-actions in ergodic theory, Progr. Math. 265 (2008), 325-351.
[3]	A. I. Danilenko, Explicit solution of Rokhlin's problem on homogeneous spectrum and applications, Ergod. Th. & Dyn. Syst. 26 (2006), 1467-1490.
[4]	A. I. Danilenko, On new spectral multiplicities for ergodic maps, Studia Math. 197 (2010), 57-68.
[5]	A. I. Danilenko and S. V. Solomko, Ergodic Abelian actions with homogeneous spectrum, Contemp. Math. 532 (2010), 137-148.
[6]	I. Filipowicz, Product $Z^d$-actions on a Lebesgue space and their applications, Studia Math. 122 (1997), 289-298.
[7]	K. Frączek, Cyclic space isomorphism of unitary operators, Studia Math. 124 (1997), 259-267.
[8]	G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5 (1999), 173-226.
[9]	E. Hewitt and K. A. Ross, "Abstract Harmonic Analysis'', Vol. I, Springer-Verlag, Berlin-Göt-tingen-Heidelberg, 1963.
[10]	A. Katok and M. Lemańczyk, Some new cases of realization of spectral multiplicity function for ergodic transformations, Fund. Math. 206 (2009), 185-215.
[11]	A. Katok and J.-P. Thouvenot, Spectral properties and combinatorial constructions in ergodic theory, in "Handbook of Dynamical Systems'', vol. 1B, Elsevier B. V., 2006, 649-743.
[12]	R. A. Konev and V. V. Ryzhikov, On spectral multiplicities ${2,\ldots,2^n}$ for totally ergodic $\mathbbZ^2$-actions,, Preprint, ().
[13]	J. Kwiatkowski jr and M. Lemańczyk, On the multiplicity function of ergodic group extensions. II, Studia Math., 116 (1995), 207-215
[14]	A. del Junco and D. Rudolph, On ergodic actions whose self-joinings are graphs, Erg. Th. & Dyn. Syst., 7 (1987), 531-557
[15]	M. Lemańczyk, Spectral theory of dynamical systems, in "Encyclopedia of Complexity and Systems Science'' Springer-Verlag, 2009, 8554-8575.
[16]	M. Lemańczyk and F. Parreau, Special flows over irrational rotations with the simple convolutions property,, preprint., ().
[17]	G. Mackey, Induced representations of locally compact groups. I, Ann. Math. 55 (1952), 101-139.
[18]	V. V. Ryzhikov, Transformations having homogeneous spectra, J. Dynam. Control Systems, 5 (1999), 145-148.
[19]	V. V. Ryzhikov, Spectral multiplicities and asymptotic operator properties of actions with an invariant measure, Mat. Sb., 200 (2009), 107-120.
[20]	R. Zimmer, Induced and amenable ergodic actions of Lie groups, Ann. Sci. Ecole Norm. Sup., 11 (1978), 407-428.

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