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Journal Articles Pure and Applied Functional Analysis Year : 2019

A uniform continuity property of the winding number of self-mappings of the circle

Abstract

Let $u: {\mathbb S}^1\to {\mathbb S}^1$. When $u$ is continuous, it has a winding number $\text{deg}\, u$, which satisfies $\text{deg}\, u=\text{deg}\, v$ if $u, v\in C^0({\mathbb S}^1 ; {\mathbb S}^1)$ and $\|u-v\|_{L^\infty}\in [0,2)$. In particular, $u\mapsto\deg u$ is uniformly continuous for the $\sup$ norm. The winding number $\deg u$ can be naturally defined, by density, when $u$ is merely VMO. For such $u$'s, the winding number $\deg$ is continuous with respect to the BMO norm. Let $p\in (1,\infty)$. In view of the above and of the embedding $W^{1/p,p}({\mathbb S}^1)\hookrightarrow$VMO, maps in $W^{1/p,p}({\mathbb S}^1; {\mathbb S}^1)$ have a well-defined winding number, continuous with respect to the $W^{1/p,p}$ norm. However, an example due to Brezis and Nirenberg yields sequences $(u_n), (v_n)\subset W^{1/p,p}({\mathbb S}^1; {\mathbb S}^1)$ such that $\|u_n-v_n\|_{W^{1/p,p}}\to 0$ as $n\to\infty$ and $\text{deg}\, u_n\neq\text{deg}\, v_n$, $\forall\, n$. Thus $\deg$ is not uniformly continuous with respect to the $W^{1/p,p}$ norm (and, a fortiori, with respect to the BMO norm). The above sequences satisfy $\|u_n\|_{W^{1/p,p}}\to\infty$, $\|v_n\|_{W^{1/p,p}}\to\infty$. We prove that a similar phenomenon cannot occur for bounded sequences. More specifically, we prove the following uniform continuity result. Given $p\in (1, \infty)$ and $M\in (0,\infty)$, there exists some $\delta=\delta(p, M)\in (0,\infty)$ such that \begin{equation*} [\|u\|_{W^{1/p,p}}\le M, \, \|u-v\|_{W^{1/p,p}}\le \delta]\implies \text{deg}\, u=\text{deg}\, v. \end{equation*}
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Dates and versions

hal-02151571 , version 1 (08-06-2019)
hal-02151571 , version 2 (02-09-2019)

Identifiers

  • HAL Id : hal-02151571 , version 2

Cite

Petru Mironescu, Itai Shafrir. A uniform continuity property of the winding number of self-mappings of the circle. Pure and Applied Functional Analysis, 2019, 5 (5), pp.1199-1204. ⟨hal-02151571v2⟩
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