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

inner ring

0.1

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

z

is a fixed point of

f

if

f(z)=z.

) The method used is to map the interior of the disk to the whole plane with

z↦z/(1-|z|)

, translate by the shift, and map back to the disk with

z↦z/(1+|z|)

.