American Institute of Mathematical Sciences

Advanced
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

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 & Continuous Dynamical Systems - A, 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), 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

show all references

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

[1]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[2]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[3]

Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks & Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453
[4]

Oliver Jenkinson. Ergodic Optimization. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 197-224. doi: 10.3934/dcds.2006.15.197
[5]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[6]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[7]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
[8]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121
[9]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[10]

Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87
[11]

Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456
[12]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[13]

Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201
[14]

Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049
[15]

Ian D. Morris. Ergodic optimization for generic continuous functions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 383-388. doi: 10.3934/dcds.2010.27.383
[16]

Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339
[17]

Andreas Strömbergsson. On the deviation of ergodic averages for horocycle flows. Journal of Modern Dynamics, 2013, 7 (2) : 291-328. doi: 10.3934/jmd.2013.7.291
[18]

O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[19]

Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459
[20]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. Ergodic control for a mean reverting inventory model. Journal of Industrial & Management Optimization, 2018, 14 (3) : 857-876. doi: 10.3934/jimo.2017079

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences