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Analytic order-isomorphisms of countable dense subsets of the unit circle

Published online by Cambridge University Press:  14 July 2021

Maxim R. Burke*
Affiliation:
School of Mathematical and Computational Sciences, University of Prince Edward Island, Charlottetown, PE C1A 4P3, Canada

Abstract

For functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$ , $1\notin B$ , then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$ . This answers a question of P. M. Gauthier.

Type
Article
Copyright
© Canadian Mathematical Society 2021

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Footnotes

Research supported by NSERC.

References

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), 620626.10.1112/jlms/2.Part_4.620CrossRefGoogle Scholar
Burke, M. R., Approximation and interpolation by entire functions with restriction of the values of the derivatives . Topol. Appl. 213(2016), 2449.10.1016/j.topol.2016.08.005CrossRefGoogle Scholar
Deutsch, F., Simultaneous interpolation and approximation in topological linear spaces . SIAM J. Appl. Math. 14(1966), 11801190.CrossRefGoogle Scholar
Rudin, W., Real and complex analysis. 3rd ed., McGraw-Hill, New York, 1986.Google Scholar
Sohrab, H., Basic real analysis. 2nd ed., Birkhäuser, New York, 2014.Google Scholar