Uniqueness of Brouwer fixed point theorem

Question

I am reading Brouwer’s fixed point thereom. Does the uniqueness holds in Brouwers theorem i.e.

Is it true that every non identity continuous map $f: D \to D$ has unique fixed point?

D- closed unit disc

Any hints or ideas?

Answer 1

No. Take the function $f:\mathbb{R}^2\to\mathbb{R}^2$ given by $f(x,y)=(x,-y)$. It maps the disk onto itself, is not the identity, and has a continuum of fixed points of the form $(x,0)$.