Recent Progress on Fourier Decay for Stationary Measures

The study of Fourier decay for stationary measures is a classical problem, with roots in Erdős’ 1939 work on Bernoulli convolutions. Over the years, tools and ideas from number theory, smooth dynamics, and arithmetic combinatorics have led to substantial progress, yet several fundamental questions remain open. After introducing the problem and its history, I will present recent joint work with Federico Rodriguez Hertz and Zhiren Wang establishing, essentially optimal results for a large class of stationary measures with respect to smooth nonlinear maps.