# Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map.

-editted- Let $x^\star$ be a fixed point of $F$.

Then, is it true that the fixed point iteration $x_{n+1} = F(x_n)$ converges locally if and only if the spectral radius of the Jacobian $J_F(x)$ at $x = x^\star$ is less than 1?

No. If you have a fixed point and the Jacobian at this fixed point has a spectral radius smaller $1$, then the fixed point is stable or attracting.
There may be many other, non-fixed, points where the Jacobian has a small spectral radius. Take $f(x)=x/2$. Then $f'(x)=0.5$ everywhere, but only $x=0$ is a fixed point.