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October  2019, 39(10): 6001-6021. doi: 10.3934/dcds.2019262

On the isomorphism problem for non-minimal transformations with discrete spectrum

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Received  December 2018 Revised  May 2019 Published  July 2019

The article addresses the isomorphism problem for non-minimal topological dynamical systems with discrete spectrum, giving a solution under appropriate topological constraints. Moreover, it is shown that trivial systems, group rotations and their products, up to factors, make up all systems with discrete spectrum. These results are then translated into corresponding results for non-ergodic measure-preserving systems with discrete spectrum.

Citation: Nikolai Edeko. On the isomorphism problem for non-minimal transformations with discrete spectrum. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6001-6021. doi: 10.3934/dcds.2019262
References:
[1]

A. Arhangel'skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, 2008. doi: 10.2991/978-94-91216-35-0.  Google Scholar

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N. Bourbaki, General Topology: Chapters 1–4, Springer-Verlag, Berlin, 1989.  Google Scholar

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J. R. Choksi, Non-ergodic transformations with discrete spectrum, Illinois J. Math., 9 (1965), 307-320.  doi: 10.1215/ijm/1256067892.  Google Scholar

[4]

J. Dugundji, Topology, Allyn and Bacon, 1966.  Google Scholar

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T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Springer, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar

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R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc., 94 (1960), 272-281.  doi: 10.1090/S0002-9947-1960-0123636-3.  Google Scholar

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E. Glasner, Enveloping semigroups in topological dynamics, Topol. Appl., 154 (2007), 2344-2363.  doi: 10.1016/j.topol.2007.03.009.  Google Scholar

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A. Gleason, Projective topological spaces, Illinois J. Math., 2 (1958), 482-489.  doi: 10.1215/ijm/1255454110.  Google Scholar

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M. Haase and N. Moriakov, On systems with quasi-discrete spectrum, Stud. Math., 241 (2018), 173-199.  doi: 10.4064/sm8756-6-2017.  Google Scholar

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P. R. Halmos and J. von Neumann, Operator methods in classical mechanics, Ⅱ, Ann. Math., 43 (1942), 332-350.  doi: 10.2307/1968872.  Google Scholar

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K. Jacobs, Ergodentheorie und fastperiodische Funktionen auf Halbgruppen, Math. Z., 64 (1956), 298-338.  doi: 10.1007/BF01166575.  Google Scholar

[12]

J. Kwiatkowski, Classification of non-ergodic dynamical systems with discrete spectra, Comment. Math., 22 (1981), 263-274.   Google Scholar

[13]

E. Michael, Continuous selections ${\rm I}$, Ann. Math., 63 (1956), 361-382.  doi: 10.2307/1969615.  Google Scholar

[14]

J. Renault, A Groupoid Approach to ${\rm C}^*$-Algebras, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1980.  Google Scholar

[15]

H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar

[16]

M. Takesaki, Theory of Operator Algebras I, Springer, 1979.  Google Scholar

[17]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932), 587-642.  doi: 10.2307/1968537.  Google Scholar

show all references

References:
[1]

A. Arhangel'skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, 2008. doi: 10.2991/978-94-91216-35-0.  Google Scholar

[2]

N. Bourbaki, General Topology: Chapters 1–4, Springer-Verlag, Berlin, 1989.  Google Scholar

[3]

J. R. Choksi, Non-ergodic transformations with discrete spectrum, Illinois J. Math., 9 (1965), 307-320.  doi: 10.1215/ijm/1256067892.  Google Scholar

[4]

J. Dugundji, Topology, Allyn and Bacon, 1966.  Google Scholar

[5]

T. Eisner, B. Farkas, M. Haase and R. Nagel, Operator Theoretic Aspects of Ergodic Theory, Springer, 2015. doi: 10.1007/978-3-319-16898-2.  Google Scholar

[6]

R. Ellis, A semigroup associated with a transformation group, Trans. Amer. Math. Soc., 94 (1960), 272-281.  doi: 10.1090/S0002-9947-1960-0123636-3.  Google Scholar

[7]

E. Glasner, Enveloping semigroups in topological dynamics, Topol. Appl., 154 (2007), 2344-2363.  doi: 10.1016/j.topol.2007.03.009.  Google Scholar

[8]

A. Gleason, Projective topological spaces, Illinois J. Math., 2 (1958), 482-489.  doi: 10.1215/ijm/1255454110.  Google Scholar

[9]

M. Haase and N. Moriakov, On systems with quasi-discrete spectrum, Stud. Math., 241 (2018), 173-199.  doi: 10.4064/sm8756-6-2017.  Google Scholar

[10]

P. R. Halmos and J. von Neumann, Operator methods in classical mechanics, Ⅱ, Ann. Math., 43 (1942), 332-350.  doi: 10.2307/1968872.  Google Scholar

[11]

K. Jacobs, Ergodentheorie und fastperiodische Funktionen auf Halbgruppen, Math. Z., 64 (1956), 298-338.  doi: 10.1007/BF01166575.  Google Scholar

[12]

J. Kwiatkowski, Classification of non-ergodic dynamical systems with discrete spectra, Comment. Math., 22 (1981), 263-274.   Google Scholar

[13]

E. Michael, Continuous selections ${\rm I}$, Ann. Math., 63 (1956), 361-382.  doi: 10.2307/1969615.  Google Scholar

[14]

J. Renault, A Groupoid Approach to ${\rm C}^*$-Algebras, vol. 793 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 1980.  Google Scholar

[15]

H. Schaefer, Banach Lattices and Positive Operators, Springer-Verlag, New York-Heidelberg, 1974.  Google Scholar

[16]

M. Takesaki, Theory of Operator Algebras I, Springer, 1979.  Google Scholar

[17]

J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik, Ann. Math., 33 (1932), 587-642.  doi: 10.2307/1968537.  Google Scholar

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