Consider a dynamical map at a fixed point (from Wikipedia):
There is an analogous criterion for a continuously differentiable map $f$: $R^n → R^n$ with a fixed point $a$, expressed in terms of its Jacobian matrix at $a$, $J_a(f)$. If all eigenvalues of $J$ are real or complex numbers with absolute value strictly less than 1 then a is a stable fixed point; if at least one of them has absolute value strictly greater than 1 then a is unstable.
If I have two maps $A, B$, each with their largest eigenvalue as $|\lambda_A| \ll |\lambda_B| < 1$, can I make any conclusions that map $A$ converges onto its fixed point faster than map $B$?
Yes, in a sense. But if you compare the convergence rates of individual points, the relation can be reverse.
Consider linear maps, identified with their matrices, \begin{equation} A = \begin{bmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{bmatrix}, \quad B = \begin{bmatrix} \lambda_3 & 0 \\ 0 & \lambda_3 \end{bmatrix}, \end{equation} where $0 < \lambda_2 <\lambda_3 < \lambda_1 < 1$. Clearly, the speed of convergence of $B^{k} x$ is $\lambda_3$, for any $x \in R^2$. Now, the speed of convergence of $A^k x$ is $\lambda_1$, but only for $x = (x_1, x_2)^{\top}$ with $x_1 \ne 0$. The speed of convergence of $A^k (0, x_2)^{\top}$ is $\lambda_2$ (which is $< \lambda_3$).
In the nonlinear case, assume that $J_a(f)$ has an eigenvalue $\overline\lambda$ of multiplicity $l$ with the largest modulus, and that the remaining eigenvalues have moduli less than some $\mu < \lvert \overline\lambda \rvert$. Then there exists, in a nbhd of $a$, a "very stable" $(n - l)$-dimensional invariant $C^1$ submanifold, tangent at $a$ to the sum of the invariant subspaces corresponding to those eigenvalues with moduli less than $\mu$. On that "very stable" manifold the convergence to $a$ is at least as strong as $\mu$, whereas the remaining points close to $a$ converge to $a$ not faster than with speed $\lvert \overline\lambda \rvert$. For a reference, see, e.g. Theorem 4.1 on p. 326 of: C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics 34, Springer, 2006.