Lyapunov exponents for products of matrices

Let \({\bf M}=(M_1,\ldots, M_k)\) be a tuple of real \(d\times d\) matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether \({\bf M}\) possesses the following property: there exist two constants \(\lambda\in {\Bbb R}\) and \(C>0\) such that for any \(n\...

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Bibliographic Details
Published in:arXiv.org 2017-02
Main Authors: De-Jun, Feng, Chiu-Hong, Lo, Shen, Shuang
Format: Article
Language:English
Subjects:
Continuity
Liapunov exponents
Self-similarity
Online Access:Get full text
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Summary:Let \({\bf M}=(M_1,\ldots, M_k)\) be a tuple of real \(d\times d\) matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether \({\bf M}\) possesses the following property: there exist two constants \(\lambda\in {\Bbb R}\) and \(C>0\) such that for any \(n\in {\Bbb N}\) and any \(i_1, \ldots, i_n \in \{1,\ldots, k\}\), either \(M_{i_1} \cdots M_{i_n}={\bf 0}\) or \(C^{-1} e^{\lambda n} \leq \| M_{i_1} \cdots M_{i_n} \| \leq C e^{\lambda n}\), where \(\|\cdot\|\) is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on \({\Bbb R}\), the absolute continuity of certain self-affine measures in \({\Bbb R}^d\) and the dimensional regularity of a class of sofic affine-invariant sets in the plane.
ISSN:2331-8422