Statistical properties of mostly expanding fast-slow partially hyperbolic systems
Jacopo De Simoi, Kasun Fernando, Nicholas Fleming-Vázquez
TL;DR
This work analyzes fast-slow, $C^4$ partially hyperbolic maps on the torus under small perturbations, proving the existence and uniqueness of a physical measure and an exponential decay of correlations for sufficiently smooth observables in the mostly expanding setting. By combining averaging theory with a novel coupling framework based on standard patches (2D centre-unstable rectangles) and a detailed invariant centre structure, the authors derive explicit lower bounds $c_\varepsilon\ge C_2\varepsilon/\log(\varepsilon^{-1})$ for the decay rate. The results extend the De Simoi–Liverani program to the mostly expanding regime, providing near-optimal decay bounds and a robust method that could generalize to more complex fast-slow partially hyperbolic systems. The approach hinges on (i) averaging the fast expanding dynamics to a one-dimensional averaged drift, (ii) constructing invariant centre foliations and a hierarchy of standard patches, and (iii) coupling patch measures to obtain memory loss and decorrelation.
Abstract
We consider a class of fast-slow $C^4$ partially hyperbolic systems on $\mathbb{T}^2$ given by $ε$-perturbations of maps $F(x,θ)=(f(x,θ),θ)$ where $f(\cdot,θ)$ are $C^{4}$ expanding maps of the circle. For sufficiently small $ε$ and an open set of perturbations we prove existence and uniqueness of a physical measure and exponential decay of correlations for sufficiently smooth observables with explicit almost optimal bounds on the decay rate. Our result complements previous work by De Simoi and Liverani, which studied the case of mostly contracting centre.