Rate of Convergence in the Weak Invariance Principle for Deterministic Systems

Commun. Math. Phys. 369 (2019) 1147-1165.

Marios Antoniou and Ian Melbourne

Abstract

We obtain the first results on convergence rates in the Prokhorov metric for the weak invariance principle (functional central limit theorem) for deterministic dynamical systems. Our results hold for uniformly expanding/hyperbolic (Axiom A) systems, as well as nonuniformly expanding/hyperbolic systems such as dispersing billiards, Hénon-like attractors, Viana maps and intermittent maps. As an application, we obtain convergence rates for deterministic homogenization in multiscale systems.

Minor typos, etc (Last updated: February 2023)

I am grateful to Nicolò Paviato for pointing out the following:

• In the third line of the proof of Lemma 4.3, it is claimed that λ₂ << λ₁. What is true (and what is used) is that the estimate for λ₂ in the previous line is no worse (up to a constant) than that for λ₁.

• Definition 6.1(b). χ should be ψ.

• Remark 6.2. "Young tails" should be "Young towers".

• In the proof of Theorem 6.3, on line 3, we can take δ=0. Indeed, the estimate on line 2 has the form |Y_n-Y_n'|_∞ ≤ Cn^-1/2 so P(|Y_n-Y_n'|_∞ ≥ 2Cn^-1/2) = 0 and hence π₁(Y_n,Y_n') ≤ 2Cn^-1/2 by Proposition 4.5(a).