# A Continuous Mapping That Fixes All Boundary Points but No Interior Points

A Continuous Mapping That Fixes All Boundary Points but No Interior Points

Except for the zero shift, each member of this family of continuous mappings of the unit disk into itself fixes the boundary pointwise but has no interior fixed points. (The point is a fixed point of if ) The method used is to map the interior of the disk to the whole plane with , translate by the shift, and map back to the disk with .

z

f

f(z)=z.

z↦z/(1-|z|)

z↦z/(1+|z|)