Spectral gaps and Fourier decay for self-conformal measures in the plane

TL;DR

This paper proves a spectral gap estimate for transfer operators of self-conformal measures in the plane and establishes polynomial Fourier decay, advancing understanding of these measures without requiring separation or Federer conditions.

Contribution

It introduces a new spectral gap estimate and Fourier decay results for self-conformal measures in the plane, relaxing previous separation and Federer assumptions.

Findings

Spectral gap estimate for transfer operators

Polynomial Fourier decay for self-conformal measures

Method refines Oh-Winter (2017) approach

Abstract

Let $\Phi$ be a $C^\omega (\mathbb{C})$ self-conformal IFS on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay for any such measure. Our technique is based on a refinement of a method of Oh-Winter (2017) where we do not require separation from the IFS or the Federer property for the underlying measure.