Logarithmic Fourier decay for self conformal measures

TL;DR

This paper proves that the Fourier transform of self conformal measures on the real line decays logarithmically at infinity unless the system is smoothly conjugated to a linear, non-Diophantine contracting IFS, using an effective local limit theorem.

Contribution

It introduces an effective local limit theorem for cocycles with moderate deviations, providing new insights into Fourier decay of self conformal measures.

Findings

Fourier transform decays logarithmically at infinity for most self conformal measures.

Identifies specific conditions where decay does not occur, related to linear conjugation and Diophantine properties.

Develops a technical result of independent interest: an effective local limit theorem for cocycles.

Abstract

We prove that the Fourier transform of a self conformal measure on $\mathbb{R}$ decays to $0$ at infinity at a logarithmic rate, unless the following holds: The underlying IFS is smoothly conjugated to an IFS that both acts linearly on its attractor and contracts by scales that are not Diophantine. Our key technical result is an effective version of a local limit Theorem for cocycles with moderate deviations due to Benoist-Quint (2016), that is of independent interest.