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 H\"older 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 $\kappa<1/4$ and $p<\infty$. Additionally, this is the first result on rates that covers certain invertible, slowly mixing maps, such as Bunimovich flowers.