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Recurrent Transformation Groups

Published online by Cambridge University Press:  20 November 2018

R. A. Christiansen
R. A. Christiansen*
Affiliation:
Grinnell College, Grinnell, Iowa
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Let (X, T, π) denote a flow, where X is a compact topological space metrizable by d, and T is a closed non-trivial subgroup of the reals under addition. T is recurrent if and only if for each and s > 0, there exists t > s such that x ∈ X implies . If T is almost-periodic, then T is both recurrent and distal. In §§ 4 and 5, it is shown that, under more stringent hypotheses, the recurrence of T is neither a necessary nor a sufficient condition for T to be distal. Let S be a closed non-trivial subgroup of T. It is shown in § 3 that T is recurrent if and only if S is recurrent. From this result, we obtain a solution to a problem posed by Nemyckiĭ (16, p. 492, Problem 6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 564 - 575
Copyright
Copyright © Canadian Mathematical Society 1969

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