Given $\mathcal A \subset R^{n \times n}$.
The joint spectral radius is by: $$\sigma( \mathcal A) = \limsup_{m \rightarrow \infty}\sup_{A \in \mathcal A^m}\rho(A),$$ where $\rho$ is the normal spectral radius an $\mathcal A^m = \{A_1 \cdot A_2\cdot \ldots \cdot A_m: A_i \in \mathcal A, i = 1, \ldots , m\}$.
I want to know, if it is possible to make a statement about the joint spectral radius $\sigma(\mathcal A)$, if we know that $\rho(A)<1 ~ \forall A \in \mathcal A$.