Vol. 37, No. 1, 1971

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Mapping solenoids onto strongly self-entwined, circle-like continua

James Ted Rogers Jr.

Vol. 37 (1971), No. 1, 213–216

A circle-like continuum C is self-entwined if there exists a sequence {Ci} of circular chains which define C, a point p in C, and a sequence {Di} such that, for each i, (1) either Di is a subchain of Ci, or Di = Ci, (2) Di+1 circles at least twice in Ci, (3) Ci+1 circles at least once in Ci, and (4) the point p is in the first link of Di. If, in addition, each Di+1 circles more times in Ci than Ci+1 circles in Ci, then C is said to be strongly self-entwined.

The purpose of this paper is to prove the following.

Theorem 1. No solenoid can be mapped onto a strongly self-entwined, circle-like continuum.

Mathematical Subject Classification
Primary: 54F20
Received: 10 October 1969
Revised: 21 September 1970
Published: 1 April 1971
James Ted Rogers Jr.