Statistical properties of mostly contracting fast-slow partially hyperbolic systems

We consider a class of $\mathcal C^{4}$ partially hyperbolic systems on $\mathbb T^2$ described by maps $F_\varepsilon(x,θ)=(f(x,θ),θ+\varepsilonω(x,θ))$ where $f(\cdot,θ)$ are expanding maps of the circle. For sufficiently small $\varepsilon$ and $ω$ generic in an open set, we precisely classify the SRB measures for $F_\varepsilon$ and their statistical properties, including exponential decay of correlation for Hölder observables with explicit and nearly optimal bounds on the decay rate.

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