Let $\mathcal H(\mathbb X)$ be a set of all compact subsets of $\mathbb X$ (it's a complete metric space). Let $\mathcal F_{\mathbf A, \mathbf b}\colon \mathcal H(\mathbb X) \to \mathcal H(\mathbb X)$ be a contraction mapping parametrized by the square matrix $\mathbf A$ and vector $\mathbf b$ of the form: $$\mathcal F_{\mathbf A, \mathbf b}(S) = \{\mathbf A \mathbf z + \mathbf b \mid \mathbf z \in S \}$$ Is there a non-linear monotonic function $f\colon \mathbb R \to \mathbb R$ s.t. $f \circ \mathcal F$ is also contraction? Here $f(S) = \{(f(z_1), \ldots, f(z_{n})) \mid (z_1, \ldots, z_{n}) \in S)\}$.
Actually, I'm interested in the case $\mathbb X = [0, 1]^{n}$ and $\mathbb X = \mathbb R^{n}$.