Constellations for Lyapunov exponents

Abstract

We study the maximal Lyapunov exponent of random products of positive matrices, and prove that it can be calculated to arbitrary precision with arithmetic work polynomial in the number of requested digits. In nondegenerate settings, the complexity of our method depends on the intrinsic complexity of the dynamics---which we characterize---and not on the Euclidean dimension of the ambient space in which the dynamics live. As numerical applications, we determine the Hausdorff dimension of random intersections of Sierpiński gaskets, and Viswanath's constant---the growth rate of the random Fibonacci sequence. We first establish a general inequality for nonnegative matrix families showing that, if a signed measure with finite support is sufficiently close to being stationary, then it predicts the Lyapunov exponent accurately. This framework also encompasses Pollicott's Fredholm determinant approximation. We then show that, just like drawing a constellation by picking bright stars, one can construct a sequence of signed measures approximating stationarity by picking informative orbits of the projective dynamics. Furthermore, we prove that our construction uses an asymptotically optimal number of atoms, which is governed by an intrinsic quantity we call constellation dimension.