Vol. 50, No. 1, 1974

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Homeomorphisms of long circles without periodic points

Richard Freiman

Vol. 50 (1974), No. 1, 47–56

Markley characterized those minimal sets arising from a homeomorphism of the circle without periodic points. The removal of the condition that guarantees the space be metric yields a larger class of minimal sets which can still be embedded in a circle-like object. A {long} {short} circle is a compact connected {non-metric Hausdorff} {metric} space which has no cut points, but which is disconnected by the removal of any two points. A circle is either a long or short circle. Theorem: A compact Hausdorff minimal cascade can be embedded in a homeomorphism of a circle without periodic points if and only if it is proximally equicontinuous over the circle and PΔ, where P is the proximal relation.

Mathematical Subject Classification 2000
Primary: 54H20
Received: 20 July 1972
Revised: 17 August 1973
Published: 1 January 1974
Richard Freiman