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$\text{PL}_{+}(I)$ is not a Polish group

Published online by Cambridge University Press:  06 October 2015

MICHAEL P. COHEN  and
ROBERT R. KALLMAN
MICHAEL P. COHEN
Affiliation:
Department of Mathematics, North Dakota State University, PO Box 6050, Fargo, ND 58108-6050, USA email michael.cohen@ndsu.edu
ROBERT R. KALLMAN
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle 311430, Denton, TX 76203-5017, USA email kallman@unt.edu
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Abstract

The group $\text{PL}_{+}(I)$ of increasing piecewise-linear self-homeomorphisms of the interval $I=[0,1]$ may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups $\text{Homeo}_{+}^{\text{Lip}}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\text{Diff}_{+}^{1+\unicode[STIX]{x1D716}}(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $\unicode[STIX]{x1D716}$, as well as the corresponding groups acting on the real line and on the circle.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 7 , October 2016 , pp. 2121 - 2137
Copyright
© Cambridge University Press, 2015 

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