Surjective map on compact metric space Is there a surjective map $f:X\to X$ on compact metric spcace $(X, d)$ with the following condition?
There is $0<L<1$ such that $d(f(x), f(y))<Ld(x, y)$  for all $x,y\in X$

Let $x, y \in X$. Since $f$ is surjective, there are $x_1, y_1 \in X$ such that $f(x_1)=x, f(y_1)=y$, similarly $f(x_2)=x_1, f(y_2)=y_1$, ..., $f(x_{n+1})=x_n, f(y_{n+1})=y_n$. From the assumption,
$$d(f(x), f(y))<Ld(x, y)=Ld(f(x_1), f(y_1))<L^2d(x_1, y_1)<\cdots <L^{n+1}d(x_{n}, y_{n})\leqslant L^{n+1} M,$$
for all $n$, where $M:=\sup_{x, y\in X}d(x, y)<\infty$. By letting $n\to \infty$, we obtain from $L^n \to 0$ that $d(f(x), f(y))=0$. Hence, $f$ is constant on $X$. This means $X$ is singleton and there has no such function.
