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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Multi-invariant sets on tori


Author: Daniel Berend
Journal: Trans. Amer. Math. Soc. 280 (1983), 509-532
MSC: Primary 11K06; Secondary 11K55, 28D10, 54A15
DOI: https://doi.org/10.1090/S0002-9947-1983-0716835-6
MathSciNet review: 716835
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Abstract: Given a compact metric group $ G$, we are interested in those semigroups $ \Sigma $ of continuous endomorphisms of $ G$, possessing the following property: The only infinite, closed, $ \Sigma $-invariant subset of $ G$ is $ G$ itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization--for the case of finitedimensional tori--of those commutative semigroups with the aforementioned property.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0716835-6
Keywords: Invariant set, finite-dimensional torus, semigroup of endomorphisms, ergodic endomorphism, minimal set
Article copyright: © Copyright 1983 American Mathematical Society