Can $x \rightarrow \sqrt{x}$ be considered as one dimensional discrete dynamical system? As I see it will have the fixed points $x^{*} = 0,1$.
I am not sure about the stability of $x^* = 0$ as $|f'(x^*)|$ is not defined where $f(x) = \sqrt{x}$. The fixed point at $x=1$ is stable as $|f'(1)|<1$. So all positive real numbers except zero will converge to the fixed point $x^* = 1$.