This question is a cross post from the Math StackExchange since it got no attention at all there: https://math.stackexchange.com/questions/4414601/maps-that-preserve-winding-numbers
I am looking for a characterisation of the continuous maps on some subset of $A\subseteq \mathbb{C}$ that preserve the winding numbers of all closed curves in $A$, i.e. if $\gamma$ is a closed curve that lies in $A$ and $x\in A$ is a point not lying on $\gamma$, then $$\text{ind}_{f\circ \gamma}(f(x)) = \text{ind}_\gamma(x)\ .$$
Translations clearly satisfy this. Multiplications with a non-zero complex number do as well. $\mathbb{R}$-linear maps with positive determinant probably as well. $\mathbb{R}$-linear maps with negative determinant on the other hand will flip the sign of the winding number.
Another example is the inverse function, which has this property on any region not containing 0.
Is there any good classification of functions that have that kind of property?
Best Answer
I found an answer to this question: the question of how a continuous function changes the winding number of a closed curve can be studied quite generally. The important concept here is the degree of the function (also known as the Brouwer degree).
If a curve $\gamma$ has winding number $n$ around a point $z$ (say w.l.o.g. $z = 0$) then it is homotopic to a circle that winds $n$ times around $z$. So if we want to know the winding number of $f(\gamma)$ around $f(0)$, it is enough to look at how $f$ acts on a simple counter-clockwise circle around $0$, since and that is its degree and we have $$\text{ind}_{f(\gamma)}(0) = \text{deg}_f(f(0)) \cdot \text{ind}_\gamma(0)$$
In fact, the idea of a degree generalises the concept of winding number.
The degree of a particular function $f$ that is real-differentiable can also be easily computed using the formula $$\text{deg}_f(f(0)) = \sum_{y\in f^{-1}(0)} \text{sgn}\ \text{det}(Df(y))$$ where $Df(y)$ is the Jacobian of $f$ at $y$.
In particular, a real-differentiable function that is one-to-one and whose Jacobian has positive determinant at $0$ preserves the winding number at $0$.
At least I think this is the basic idea; I am not 100% clear on the details here, so I apologise for any mistakes in this answer. But perhaps this already helps someone who's struggling like I was!