Abstract: We prove polynomial Fourier decay for self conformal measures w.r.t a real C^2 smooth IFS, under mild non-linearity assumptions. One of the key ingredients in our argument is a cocycle version of Dolgopyat's method that is used to study the transfer operator.  In particular, our version of Dolgopyat's method does not require the cylinder covering of the underlying fractal to be a Markov partition.
Joint with Federico Rodriguez Hertz and Zhiren Wang.