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

James Ted Rogers Jr.

Vol. 37 (1971), No. 1, 213–216

DOI: 10.2140/pjm.1971.37.213

Abstract

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

Milestones

Received: 10 October 1969

Revised: 21 September 1970

Published: 1 April 1971