Effective estimates of Lyapunov exponents for random products of positive matrices

Abstract

In this note we describe estimates on the error when calculating the Lyaponov exponent for random products of positive matrices using dynamical determinants. This extends the results in (Jurga N and Morris I 2019 Nonlinearity 32 4117–46; Pollicott M 2010 Invent. Math. 181 209–26) by drawing upon a new approach introduced in (Jenkinson O and Pollicott M 2018 Adv. Math. 325 87–115; Jenkinson O, Pollicott M and Vytnova P 2018 J. Stat. Phys. 170 221–53).

Average-case complexity of the Whitehead problem for free groups

“…To summarize, our upper bounds are sometimes better, sometimes not, compared to those of [10] and [22]. At the same time, our combinatorial method is much simpler than analytical methods of [10], [19], and [22].…”

“…This gives an easily computable upper bound on λ. We note that in [19] and [22], upper and lower bounds on λ were obtained using altogether different techniques, with [22] specifically addressing the case where A = A(k), B = B(m) (in the notation of our Section 2.1).…”

“…We touch upon this problem in Section 4, although our theoretical contribution in this direction is rather modest: for matrices with nonnegative entries, our method in Section 3 provides an upper bound for the Lyapunov exponent, see Section 4.1. We note that altogether different (analytical) methods for bounding the Lyapunov exponent were used in [10], [19] and [22].…”