Rates in almost sure invariance principle for slowly mixing dynamical systems

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau-Manneville intermittent maps, with H"older continuous observables. Our rates have form $o(n^{\gamma }L(n))$, where $L(n)$ is a slowly varying function and $\gamma$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1\mathrm{/}4})$. To break the $O(n^{1\mathrm{/}4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes-Liu-Wu and Cuny-Dedecker-Merlev`ede on the Koml'os-Major-Tusn'ady approximation for dependent processes.