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:

Pelikan, S., Slaminka, E. (1987). A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds. Ergodic Theory and Dynamical Systems, 7(3), 463-479. https://doi.org/10.1017/S0143385700004132

In the case of $C^1$ diffeomorphisms there is an earlier version (with much easier proof) here:

Simon, Carl P. "A Bound for the Fixed-Point Index of an Area-Preserving Map with Applications to Mechanics.." Inventiones mathematicae 26 (1974): 187-200.


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