Let $\theta: \mathbb{N} \rightarrow \Sigma$ a switching signal, $\Sigma=\{1,\dots,m\}$ where $m$ is an integer and $ m \ge 2$, and let $\mathcal A=\{A_\sigma \in \mathbb{R}^{n\times n} |\sigma \in \Sigma\}$ be a set of invertible matrices, we denote by $\mathcal S(\Sigma)$ the set of switching signals, consider a switching signal $\theta \in \mathcal S(\Sigma) $, for $T \ge 1$, we define:$${\prod_{0}^{T-1}} A_{\theta(t)} = A_{\theta(T-1)} \times \dots \times A_{\theta(0)}$$ $\rho(\mathcal{A})= \lim_{T\rightarrow +\infty} \left( \sup_{\theta \in \mathcal S(\Sigma)}\|\prod_{0}^{T-1}A_{\theta(t)} \|^{1/T} \right) $ is the joint spectral radius of $\mathcal A$ where $\|.\|$ is some matrix norm.
The goal is to show that for all $\rho > \rho(\mathcal{A})$, there exists $C >0$ such that \begin{equation} \label{eq:traj} \forall \theta\in \mathcal S(\Sigma),\,\; \forall T\in \mathbb N,\; \|{\prod_{0}^{T-1}} A_{\theta(t)}\| \le C \rho^T . \end{equation} I don't know how to proceed, any hints?
First note that $\rho(\mathcal{A}) \geq \rho(A_\sigma), \forall \sigma \in \Sigma$. This is because $\theta(\sigma) = \{\sigma, \sigma, \dots \}$ (i.e. no switch at all) is a valid choice of a switching signal and it is well-known that $\rho(A) = \lim_{k \to \infty} \lVert A^k \rVert^{1/k}$ for any matrix. Also, there exists a $C_A > 0$ such that $C_A \rho(A) \geq \lVert A \rVert$ for any matrix. So, $$ \left\lVert \prod_{t=0}^{T-1} A_{\theta(t)} \right\rVert \leq \prod_{t=0}^{T-1} \lVert A_{\theta(t)} \rVert \leq \prod_{t=0}^{T-1} C_{A_{\theta(t)}} \prod_{t=0}^{T-1} \rho(A_{\theta(t)}) \leq C \rho(\mathcal{A})^T \leq C \rho^T $$