Fourier transforms and iterated function systems
arXiv:2311.00585
Abstract
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.
To appear in proceedings "Recent Developments in Fractals and Related Fields, Conference on Fractals and Related Fields IV, Ile de Porquerolles, France, 2022". v3: fixed typos, updated references and made various fixes suggested by the referee