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

TL;DR

This paper establishes optimal moment bounds for Birkhoff sums and iterated sums in nonuniformly hyperbolic maps modeled by Young towers with polynomial tails, extending previous results to a broader range of tail decay rates.

Contribution

It introduces a novel approach using functional correlation bounds and weak dependence to derive optimal moment bounds for a wider class of hyperbolic maps.

Findings

Proves optimal moment bounds for maps with tail decay rate  n^{-eta} for  > 2

Extends iterated moment bounds to  > 2, improving previous  > 5 results

Provides tools for applying rough path theory to homogenization in dynamical systems

Abstract

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. $