$\text{PL}_{+}(I)$ is not a Polish group. (6th October 2015)

• Record Type: Journal Article
• Authors: COHEN, MICHAEL P.; KALLMAN, ROBERT R.
• Is Part Of: Ergodic theory and dynamical systems. Volume 36:Number 7(2016)
• Page Start: 2121
• Page End: 2137
• Publication Date: 2015-10-06
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2015.13
• Languages: English
• ISSNs: 0143-3857

Abstract

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.