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

Abstract

Consider a nonuniformly hyperbolic map T:Mrightarrow M 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}vcirc T^i and iterated sums sum _{0le i<j<n}vcirc T^i, wcirc T^j , where v,w:Mrightarrow {{mathbb {R}}} are (dynamically) Hölder 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 to an Itô diffusion for fast-slow systems of the form xk+1(n)=xk(n)+n-1a(xk(n),yk)+n-1/2b(xk(n),yk),yk+1=Tyk\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \\quad y_{k+1}=Ty_k \\end{aligned}$$\\end{document}in the optimal range beta >2.