Refinement of estimates for the Lyapunov exponents of a class of linear nonautonomous systems of difference equations

We obtain an estimate for the norm of an $n$th-order square matrix $A^{t}$:
$$ \|A^{t}\|\leq \sum^{n-1}_{k=0}C^{k}_{t}\gamma^{t-k}(\gamma+\|A\|)^{k},\quad t\geq n-1, $$
where $C^{k}_{t}$ are the binomial coefficients, $\gamma=\max\limits_{i}|\lambda_{i}|$, and $\lambda_{i}$ are the eigenvalues of $A$. Based on this estimate and using the freezing method, we improve the constants in the upper and lower estimates for the highest and lowest exponents, respectively, of the system $ x(t+1)=A(t)x(t),\ x\in \mathbb R^{n},\ t\in \mathbb Z^{+}, $ with a completely bounded matrix $A(t)$. It is assumed that the matrices $A(t)$ and $A^{-1} (t)$ satisfy the inequalities $ \|A(t)-A(s)\|\leq\delta|t-s|^{\alpha},\ \|A^{-1}(t)-A^{-1}(s)\|\leq\delta|t-s|^{\alpha} $ with some constants $0<\alpha\leq 1$ and $\delta>0$ for any $t,s\in\mathbb Z^{+}$. We give an example showing that the constants $\gamma$ and $\delta$ are generally related.