Pointwise normality and Fourier decay for self-conformal measures

Abstract

Let Φ be a C1+γ smooth IFS on R, where γ>0. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points x that are absolutely normal. That is, for every integer p≥2 the sequence {pkx}k∈N equidistributes modulo 1. We thus extend several state of the art results of Hochman and Shmerkin [29] about the prevalence of normal numbers in fractals. When Φ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless Φ has an explicit structure that is associated with some integer n≥2. These conditions on the derivative cocycle are also shown to imply that every self conformal measure is a Rajchman measure, that is, its Fourier transform decays to 0 at infinity. When Φ is self similar and satisfies a certain Diophantine condition, we establish a logarithmic rate of decay.