# Winding Number and Area Preserving Maps

It is known that winding numbers can be used to prove the existence of fixed points, see the following question Winding Numbers and Fixed Point Theorems. Every fixed point can have a different index, thus a winding number of two might either correspond to two fixed points each with an index of one or to a single fixed point with an index of two. Does anyone know whether an area preserving map (which is also a homeomorphism) can ever have a fixed point with the absolute value of the index greater than 1? Thanks.

Yes; it is possible to have fixed points of an arbitrary negative index. To get an example, think first of the index $-1$ fixed point at the origin for the map $(x,y) \mapsto (2x, y/2)$. This fixed point is hyperbolic and its local dynamics consists of four "hyperbolic sectors". You can make examples with more than 4 hyperbolic sectors, to get fixed points of other negative indices. For instance the fixed point in this picture has index $-2$ (and can be made area-preserving). In general if you have a similar dynamics with $2k$ sectors the index will be $1-k$.
So the absolute value of the index may be anything. What is true however is that the index itself (for an area-preserving homeomorphism) can't be greater than $1$. This is proved here:
In the case of $C^1$ diffeomorphisms there is an earlier version (with much easier proof) here: