Statistical properties of mostly expanding fast-slow partially hyperbolic systems

We consider a class of fast-slow $C^4$ partially hyperbolic systems on ${\mathbb{T}}^2$ given by $\epsilon$-perturbations of maps $F(x,\theta )=(f(x,\theta ),\theta )$ where $f(\cdot ,\theta )$ are $C^4$ expanding maps of the circle. For sufficiently small $\epsilon$ 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.