Effective estimates on the top Lyapunov exponents for random matrix products
Abstract
We study the top Lyapunov exponents of random products of positive [ image ] matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott (2010 Inventiones Math. 181 20926), the algorithm is based on the Fredholm theory of determinants of traceclass linear operators. In this article we obtain a simpler expression for the approximations which only require calculation of the eigenvalues of finite matrix products and not the eigenvectors. Moreover, we obtain effective bounds on the error term in terms of two explicit constants: a constant which describes how far the set of matrices are from all being column stochastic, and a constant which measures the minimal amount of projective contraction of the positive quadrant under the action of the matrices.
 Publication:

Nonlinearity
 Pub Date:
 November 2019
 DOI:
 10.1088/13616544/ab31d1
 arXiv:
 arXiv:1901.10944
 Bibcode:
 2019Nonli..32.4117J
 Keywords:

 Lyapunov exponent;
 random matrix products;
 transfer operator;
 Mathematics  Dynamical Systems
 EPrint:
 Updated definition of constant r, updated examples section, new section on scope for generalising results to higher dimensions, proof of lemma 2.1 corrected