Estimating invariant measures and Lyapunov exponents

Abstract

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius-Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

Ulam's method for random interval maps

“…One may calculate that α = α 1 ≈ 0.325 12 and that β = β 1 ≈ 0.877 56, so that α + β /2 ≈ 0.763 90. Once again, we see that the Perron-Frobenius operatorP is a contraction when restricted toBV 0 (see lemma 6.2 for the definition), (5) and (6) respectively. Then…”

“…For f ∈ BV 0 , one has f 1 ( 1 2 ) var f (see lemma 4 [5], for example), so that Pf = var Pf and f = var f . Now:…”

“…as required; the inequality (52) follows as in the proof of lemma 4 [5], for example. In the case of general Lasota-Yorke maps, the result follows by noting that (52)…”

An elementary way to rigorously estimate convergence to equilibrium and escape rates

“…Remark 3. We remark that our approach being based on a vector inequality has some similarity with the technique proposed in [15]. The first inequality used is the same in both approaches, the second is different.…”

“…It is important both to have a certified quantitative estimation for this convergence at a given time (numerical purposes, rigorous computation of the invariant measure as in [14], [15], [17], see also Remark 4), or an estimation for its asymptotic speed of convergence (computer assisted proofs of the speed of decay of correlations and its statistical consequences).…”