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Fixed points on rotating continua

Published online by Cambridge University Press:  24 October 2008

E. R. Reifenberg
E. R. Reifenberg
Affiliation:
Trinity College Cambridge
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Extract

Suppose I is a bounded plane continuum whose complement is a single domain (I) and that is a (1–1) bicontinuous transformation of the plane onto itself which leaves I invariant. Cartwright and Littlewood(1) have proved the following theorem:

Theorem A. Suppose that (I) has no prime end fixed under and the frontier of I contains a fixed point P. If is any prime end of (I) containing P, and C is any curve in (I) converging to such that PeC¯, the closure of C, then for some‡ integer N the continuum I(FN) consisting of

together with the sum B(FN) of its interior complementary domains contains all fixed points in I.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 1 , January 1954 , pp. 1 - 7
Copyright
Copyright © Cambridge Philosophical Society 1954

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References

REFERENCES

(1) Cartwright, M. L. and Littlewood, J. E. Some fixed point theorems. Ann. Math., Princeton, (2), 54 (1951), 1–37.Google Scholar
(2) Newman, M. A. Topology of plane sets (Cambridge, 1939).Google Scholar
(3) Reifenberg, E. R. A separationtheorem for finite sets of plane continua. Proc. Camb. phil. Soc. 49 (1953), 573.Google Scholar