Rates of convergence in the multivariate weak invariance principle for nonuniformly hyperbolic maps

We obtain rates of convergence in the weak invariance principle (functional central limit theorem) for ${\mathbb{R}}^d$-valued Hölder observables of nonuniformly hyperbolic maps. In particular, for maps modelled by a Young tower with superpolynomial tails (e.g.\ the Sinai billiard map, and Axiom A diffeomorphisms) we obtain a rate of $O(n^{-\kappa })$ in the Wasserstein $p$-metric for all $\kappa <1\mathrm{/}4$ and $p<\mathrm{\infty }$. Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.