Effective estimates on the top Lyapunov exponents for random matrix products

Abstract

We study the top Lyapunov exponents of random products of positive 2 × 2 matrices and obtain an efficient algorithm for its computation. As in the earlier work of Pollicott [16], the algorithm is based on the Fredholm theory of determinants of trace-class 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.

A branching process model for dormancy and seed banks in randomly fluctuating environments

“…For i.i.d. sequences Pollicott ( 2010 ) and Jurga and Morris ( 2019 ) established efficient approximation schemes with super-exponential convergence rates, see also Protasov and Jungers ( 2013 ) for further bounds.…”

Fast approximation of Lyapunov exponents: Beyond the locally constant case

“…Both of these terms are O f B(U ) κ m + e −cn 1/d . Notice that by Proposition 4.6 (4) we have that equation (8) is…”

“…In this section we will make the connection between the operators L β and the Lyapunov exponent for the cocycle generated by A over the Gibbs state µ ϕ and produce a formula for the Lyapunov exponent. The general method is the same as [15], [8] so we will describe it somewhat briefly. The connection arises from the following fact, if ρ(L 0 ) = 1…”

“…In 1973 Kingman described the problem of computing Lyapunov exponents as having the pride of place among the unsolved problems of subadditive ergodic theory [9]. While in the intervening years there have been significant results making progress on estimating Lyapunov exponents for random products of positive matrices ( [15], [8], [17]) a complete and general theory remains elusive. One direction for which relatively little progress has been made is on computing Lyapunov exponents for cocycles generated by functions which are not locally constant.…”

Approximating integrals with respect to stationary probability measures of iterated function systems

“…Note that while our approximations are non-effective, in some special cases the constants C, λ appearing in (18) can be made explicit. For example, when each φ i takes the form of a linear fractional transformation, that is, φ i (x) = (a i x + b i )/(c i x + d i ) for some constants a i , b i , c i , d i (which in particular includes the important class of self-similar iterated function systems), the constants C and λ can be bounded similarly to [16,17]. Also note that since the time taken to process n steps of the algorithm is exponential in n, the error decreases super-polynomially fast in time.…”