  We discuss the problem of bounding the Fourier transforms of stationary
measures of iterated function systems (IFSs) and how the pseudo-randomness of
the IFS either due to arithmetic, algebraic or geometric reasons is reflected
in the behaviour of the Fourier transform. We outline various methods that have
been built to estimate the Fourier transform of stationary measures arising
e.g. from thermodynamical formalism, additive combinatorics, random walks on
groups and hyperbolic dynamics. Open problems, prospects and recent links to
quantum chaos are also highlighted.
