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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 and Continuous Dynamical Systems, 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

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