We prove that the Fourier transform of a self conformal measure on (Formula presented.) 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.
Bibliographical noteFunding Information:
The authors thank Tuomas Sahlsten and Kasun Fernando for some helpful discussions. The authors are also grateful to the anonymous referee for a very thorough reading and many helpful suggestions which greatly improved the presentation of the paper
© 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
ASJC Scopus subject areas
- Mathematics (all)