Fourier Decay of SRB Measures for Skew Products

We study Fourier decay of natural invariant measures for real analytic skew products $T(x,y)=(f(x),g(x,y))$ on ${\mathbb{S}}^1\times [-1,1]$, where $f$ is expanding and $0<\mathrm{\mid }{\mathrm{\partial }}_{y}g(x,y)\mathrm{\mid }<1$. Let $\mathcal{S}$ denote the associated solenoidal attractor. We prove a dichotomy: either $\mathcal{S}$ is a single horizontal graph, or both the SRB measure and the measure of maximal entropy have polynomial Fourier decay. No transversality assumption is imposed.