Relation between the spectrum of operators and Lyapunov exponents (Q1267361)

Suppose \((A_n)_{n\in \mathbb{N}}\) is a sequence of independent identically distributed real invertible \(d\times d\)-matrices with \(\| A \|^{ \alpha_0}\) and \(\| A^{-1} \|^{\alpha_0}\) integrable for some \(\alpha_0>0\). Denote by \(L_\alpha\) the sequences \((v_k)_{k\in \mathbb{Z}}\) of \(\mathbb{C}^d\)-valued random variables with \(\sup_{k\in \mathbb{Z}} E| v_k |^\alpha <\infty\), and let \(L\) be the union of the \(L_\alpha\), \(0<\alpha \leq\alpha_0\), equipped with a particular weak topology. For \(v\in L\) define \(Tv\) by \((Tv)_{k+1} =A_kv_k\), \(k \in \mathbb{Z}\), so \(T:L \to L\), and \(T\) is linear and continuous. The main theorem asserts that if \(\lambda\in\sigma(T)\), the spectrum of \(T\), then \(\log | \lambda |\) is one of the Lyapunov exponents associated with (the distribution of) \(A_n\).