As we know, the convergence criterion of scalar fixed point iteration, e.g. $x^{k+1}=f(x^k)$, is that $|f^{\prime}(x^*)|<1$.
For the vector variable version, it has been proved in the case when $f$ is an affine function, i.e. $\textbf{x}^{k+1}=f(\textbf{x}^{k})=\textbf{A}\textbf{x}^k+\textbf{b}$, the criterion is that $\rho(\textbf{A})<1$, where $\rho(\textbf{A})$ is the spectral radius (maximum eigenvalue) of $\textbf{A}$.
Here comes the question, i.e., when $f$ is not an affine function, what is the convergence criterion for such fixed point iteration?
I have read something saying that it is about $||J_f(\textbf{x})||<\rho<1$, where the norm is some natural matrix norm. However, I don't know if the norm can be any norm or some specific norm. Or there is other interpretation?