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

Abstract

AbstractConsider a nonuniformly hyperbolic map $$ T:M\rightarrow M $$ T : M → M modelled by a Young tower with tails of the form $$ O(n^{-\beta }) $$ O ( n - β ) , $$ \beta >2 $$ β > 2 . We prove optimal moment bounds for Birkhoff sums $$ \sum _{i=0}^{n-1}v\circ T^i $$ ∑ i = 0 n - 1 v ∘ T i and iterated sums $$ \sum _{0\le i<j<n}v\circ T^i\, w\circ T^j $$ ∑ 0 ≤ i < j < n v ∘ T i w ∘ T j , where $$ v,w:M\rightarrow {{\mathbb {R}}} $$ v , w : M → R are (dynamically) Hölder observables. Previously iterated moment bounds were only known for $$ \beta >5$$ β > 5 . Our method of proof is as follows; (i) prove that $$ T$$ 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 $$\begin{aligned} x^{(n)}_{k+1}=x_k^{(n)}+n^{-1}a(x_k^{(n)},y_k)+n^{-1/2}b(x_k^{(n)},y_k) , \quad y_{k+1}=Ty_k \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) , y k ) + n - 1 / 2 b ( x k ( n ) , y k ) , y k + 1 = T y k in the optimal range $$ \beta >2$$ β > 2 .