Fast slow

Martingale-coboundary decomposition for families of dynamical systems

Annales de l'Institut Henri Poincaré / Analyse Non Lineaire 35 (2018) 859-885.

Alexey Korepanov, Zemer Kosloff and Ian Melbourne

Abstract

We prove statistical limit laws for sequences of Birkhoff sums of the type Σ_0≤j≤n-1v_n o T_n^j where T_n is a family of nonuniformly hyperbolic transformations.

The key ingredient is a new martingale-coboundary decomposition for nonuniformly hyperbolic transformations which is useful already in the case when the family T_n is replaced by a fixed transformation T, and which is particularly effective in the case when T_n varies with n.

In addition to uniformly expanding/hyperbolic dynamical systems, our results include cases where the family T_n consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters), Viana maps, and externally forced dispersing billiards.

As an application, we prove a homogenization result for discrete fast-slow systems where the fast dynamics is generated by a family of nonuniformly hyperbolic transformations.

pdf file

Minor typos, etc:

In Section 7.1, need an additional assumption that a₀ is Hölder in y for every fixed x. (This ensures in the proof of Proposition 7.5 that β_x,ε is Hölder.)

Example 4.10. ce^bn should be ce^bk.

Corollary 3.2 is stated suboptimally in L^p. It holds in L^2(p-1) as mentioned in Proposition 3.5 of Antoniou & Melbourne.