Winding number for Lipschitz curves

Question

For a closed curve $\Gamma: [0,1] \to \mathbb{C}$ (i.e. $\Gamma(0) = \Gamma(1)$) let $$ \iota_\Gamma(z) := \frac{1}{2\pi i} \int_\Gamma \frac{dw}{z-w} = \frac{1}{2\pi i} \int_0^1 \frac{\Gamma'(t)}{z-\Gamma(t)}dt $$ denote its winding number around $z \in \mathbb{C}$.

We know very well, that this is well defined not only for piecewise continously differentiable curves but also for Lipschitz curves (which are, of course, only absolutely continuous and hence only almost everywhere differentiable). By an exercise in Alfohrs' Complex Analysis book the notion of the winding number can even be extended to arbitrary continuous loops.

Apart from that the standard literature only seems to mention further properties of the winding number for piecewise differentiable curves. I wonder which results still hold for Lipschitz curves.

1. A standard result is, that the winding number takes integer values and is constant in each connected component of $\mathbb{C}\setminus im(\Gamma)$. (A proof for piecewise continuously differentiable functions can be found in Rudin's Complex Analysis, p. 203. I don't even see, why this argument wouldn't extend to Lipschitz loops, as well.)
2. In general, I wonder about the continuity of $\Gamma \mapsto \iota_\Gamma(z)$ for fixed $z$ with respect to the sup norm. It seems intuitive to me, but I can't find a proof in the literature and was not able to prove it myself.

Are there further properties of the winding number for Lipschitz loops? Can you point me to literature that covers winding number for Lipschitz or more general weakly differentiable (in the Sobolev sense) curves?

Answer 1

First of all, we are talking about $\Gamma:[0,1]\to\mathbb C\setminus\{z\}$, loops not passing through $0$.

The winding number, originally defined on smooth loops, is shown to be integer-valued and continuous (hence, locally constant) with respect to uniform norm. This is what allows us to extend it to continuous paths $\Gamma$, "by continuity", as we always extend a function from a dense subset to the entire space: $$\iota_\Gamma(z) =\lim_{n\to\infty }\iota_{\Gamma_n}(z)\tag1$$ where $\Gamma_n$ are smooth and converge to $\Gamma$ uniformly. Since $\iota_{\Gamma_n}(z)$ is an integer, formula (1) really means that $\iota_{\Gamma_n}(z)$ is constant for all sufficiently large $n$ (when $\Gamma_n$ is in the homotopy class of $\Gamma$), and this value is assigned to $\iota_\Gamma(z)$. Of course, $\iota_\Gamma(z)$ is an integer too.

To find these and relevant results in the literature, use (topological) degree of a map as a key term, rather than winding number. The degree $\deg (f,\Omega,p)$ is defined for continuous maps $f:\Omega\to\mathbb R^n$ such that $p\notin f(\partial \Omega)$. Its homotopy invariance property implies that $\deg (f,\Omega,p)$ depends only on the boundary values of $f$. Given $f:S^{n-1}\to\mathbb R^n\setminus\{p\}$, one can extend it to $F:\overline{B^n}\to\mathbb R^n $ by the Tietze extension theorem; the number $\deg(F, \overline{B^n},p)$ gives a definition of the winding number of $f$ about $p$.

The beginning of the survey Fixed point theory and nonlinear problems by Browder lists the main properties of the degree, without proofs but with references. For a detailed treatment, see the books on the subject:

1. Nonlinear Functional Analysis by Deimling: Chapter 1 is a thorough treatment of the topological degree in finite dimensions. Very well written.
2. Degree Theory by Lloyd, short classical book entirely devoted to the subject of topological degree of maps. Deimling's Chapter 1 is a reworked exposition of largely the same material.
3. Degree Theory in Analysis and Applications by Fonseca and Gangbo: emphasis on recent results for degree of Sobolev maps.

The book by Fonseca and Gangbo is pricey, but the survey Sobolev maps on manifolds: degree, approximation, lifting by Petru Mironescu seems to be freely available.