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Entire functions mapping countable dense subsets of the reals onto each other monotonically

Published online by Cambridge University Press:  17 April 2009

Daihachiro Sato  and
Stuart Rankin
Daihachiro Sato
Affiliation:
Department of Mathematics, University of Saskatchewan, Regina Campus, Regina, Saskatchewan, Canada;
Stuart Rankin
Affiliation:
Department of Mathematics, University of Western Ontario, London, Ontario, Canada.
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It is shown that for arbitrary countable dense ssets A and B of the real line, there exists a transcendental entire function whose restriction to the real line is a real-valued strictly monotone increasing surjection taking A onto B The technique used is a modification of the procedure Maurer used to show that for countable dense subsets A and B of the plane, there exists a transcendental entire function whose restriction to A is a bijection from A to B.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 1 , February 1974 , pp. 67 - 70
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Barth, K.F. and Schneider., W.J., “Entire functions mapping countable dense subsets of the reals onto each other monotonically”, J. London Math. Soc. (2) 2 (1970), 620–626.CrossRefGoogle Scholar
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[5]Sato, Daihachiro and Rankin, Stuart, “Highly conformal equivalence of countable dense sets”, submitted.Google Scholar