Fourier decay in parabolic $C^{1+\alpha}$ systems with overlaps

TL;DR

This paper proves power Fourier decay for equilibrium states of certain parabolic iterated function systems with overlaps, advancing understanding of measure regularity and Fourier dimension in complex dynamical systems.

Contribution

It introduces a novel multiscale induction method combining Bourgain-Dyatlov strategy to establish Fourier decay without spectral gaps, applicable to various parabolic systems.

Findings

Proves Fourier decay for equilibrium states in parabolic IFS with overlaps.

Extends results to Patterson-Sullivan measures for hyperbolic surfaces.

Constructs examples of $C^{1+eta}$ IFSs with positive Fourier dimension not conjugate to linear systems.

Abstract

We establish power Fourier decay for equilibrium states of parabolic $C^{1+\alpha}$ iterated function systems with overlaps satisfying a multiscale nonlinearity condition. This class includes the Lyons conductance measures $\nu_t$, $0<t<1$, associated to Galton-Watson trees with equal weights yielding advance towards a conjecture of Lyons on the absolute continuity of $\nu_t$ for small $t$. Further applications include Patterson-Sullivan measures for cusped hyperbolic surfaces, extending the work of Bourgain and Dyatlov to parabolic settings, conformal measures for Manneville-Pommeau and Lorenz-type maps, and the construction of the first genuinely $C^{1+\alpha}$ IFSs whose attractors have positive Fourier dimension but are not $C^1$-conjugate to linear IFSs. The proof combines the Bourgain-Dyatlov sum-product strategy with a multiscale induction approach that bypasses the use of spectral gaps for twisted transfer operators needed in several other works in the area.