In a previous paper (Aston, P. J. & Dellnitz, M. 1999 Comput. Meth. Appl. Mech. Engng 170, 223-237) we introduced a new method for computing the dominant Lyapunov exponent of a chaotic map by using spatial integration involving a matrix norm. We conjectured that this sequence of integrals decayed proportional to 1/n. We now prove this conjecture and derive a bound on the next term in the asymptotic expansion of the terms in the sequence. The Hénon map and a system of coupled Duffing oscillators are explored in detail in the light of these theoretical results.