On the decay of the Fourier transform of self-conformal measures
Amir Algom, Federico Rodriguez Hertz, Zhiren Wang · arXiv (Cornell University) · 2020
Let $\Phi$ be a $C^{1+\gamma}$ smooth IFS on an interval $J\subset \mathbb{R}$, where $\gamma>0$. We provide mild conditions on the derivative cocycle that ensure that all non-atomic self conformal measures are Rajchman measures, that is, their Fourier transform decays to $0$ at infinity. This allows us to give many new examples of self conformal Rajchman measures, and also provides a unified proof to several pre-existing results. For example, we show that if $\Phi$ is $C^\omega$ and admits a non-atomic non-Rajchman self conformal measure, then it is $C^\omega$ conjugate to a self similar IFS satisfying: $$\text{ There exist } t,r \text{ such that its contractions } \lbrace r_1,...,r_n \rbrace \subset \lbrace t\cdot r^k: k\in \mathbb{Z}\rbrace$$ This is closely related to the work of Bourgain-Dyatlov. We also prove that if $\Phi$ is self similar and does have not this form, then any $C^{1+\gamma}$ smooth image of a non atomic self similar measure is Rajchman, assuming the derivative never vanishes. This complements a classical Theorem of Kaufman about homogeneous IFS's, and extends in many cases recent results of Li-Sahlsten about the Rajchman property in the presence of independent $r_i,r_j$. The proof relies on a version of the local limit Theorem for the derivative cocycle, that is adapted from the work of Benoist-Quint.