Polynomial Fourier decay and a cocycle version of Dolgopyat's method for self conformal measures

We show that every self conformal measure with respect to a $C^2(\mathbb{R})$ IFS $\mathrm{\Phi }$ has polynomial Fourier decay under some mild and natural non-linearity conditions. In particular, every such measure has polynomial decay if $\mathrm{\Phi }$ is $C^{\omega }(\mathbb{R})$ and contains a non-affine map. A key ingredient in our argument is a cocycle version of Dolgopyat’s method, that does not require the cylinder covering of the attractor to be a Markov partition. It is used to obtain spectral gap-type estimates for the transfer operator, which in turn imply a renewal theorem with an exponential error term in the spirit of Li (2022).