Functional Correlation Bounds and Optimal Iterated Moment Bounds for Slowly-mixing Nonuniformly Hyperbolic Maps

Consider a nonuniformly hyperbolic map $T$ modelled by a Young tower with tails of the form $O(n^{-\beta})$, $\beta>2$. We prove optimal moment bounds for Birkhoff sums $\sum_{i=0}^{n-1}v\circ T^i$ and iterated sums $\sum_{0\le i<j<n}v\circ T^i\, w\circ T^j$, where $v,w:M\to \Bbb{R}$ are (dynamically) H\"older observables. Previously iterated moment bounds were only known for $\beta>5$. Our method of proof is as follows; (i) prove that $T$ satisfies an abstract functional correlation bound, (ii) use a weak dependence argument to show that the functional correlation bound implies moment estimates. Such iterated moment bounds arise when using rough path theory to prove deterministic homogenisation results. Indeed, by a recent result of Chevyrev, Friz, Korepanov, Melbourne & Zhang we have convergence an It\^o diffusion for fast-slow systems of the form $$x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad y_{k+1}=T y_k$$ in the optimal range $\beta>2.$