Relation between the spectrum of operators and Lyapunov exponents
type: Item
instance of: scholarly article
title: Relation between the spectrum of operators and Lyapunov exponents
zbMATH Open document ID: 0912.60009
author: Nguyen Huu Du
author: Phan Le Na
published in: Acta Mathematica Vietnamica
publication date: 1998-12-15T00:00:00Z
review text: 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) \(A_n\).
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 47B99
zbMATH DE Number: 1208048
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: mean exponents
zbMATH Keywords: Oseledets flag
zbMATH Keywords: Furstenberg-Kifer flag
reviewed by: Hans Crauel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1208048