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.
- 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.)
- 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?