Examples of contraction homeomorphism

Question

Let $X$ be a compact metric space with metric $d$ and let $f:X \rightarrow X$ be a homeomorphism. We say that $f$ is a contraction, if there exists $0 < c < 1$ such that $d(f(x), f(y)) \leq c d(x, y) $ for every pair of distinct points $x, y\in X$. Will you please help me to find some examples of such maps (Please ignore matrix based maps).

Answer 1

There is no such map. Let $x,x'\in X$ such that $d(x,x')$ is maximal. Take, if possible, $y,y'\in X$ such that $f(y)=x$ and that $f(y')=x'$. Then$$d(x,x')=d\bigl(f(y),f(y')\bigr)\leqslant cd(y,y')<d(y,y')\leqslant d(x,x')$$which is impossible. So, no such $y,y'\in X$ exist and therefore $f$ is not a homeomorphism.