Fourier Dimension at $C^{1+α}$ Regularity

TL;DR

The paper proves power-law Fourier decay for equilibrium states of genuinely $C^{1+\alpha}$ IFSs by introducing the Measure Non-Linearity (MNL) criterion and an induction-on-scales framework to verify a multiscale non-concentration condition (QNL). It provides two complementary routes: (i) in the rough $C^{1+\alpha}$ setting, MNL ⇒ QNL for finite-branch IFSs and a fixed-point construction in a moduli space to produce explicit $C^{1+\alpha}$ examples with decay; (ii) in smoother regimes, Uniform Non-Integrability (UNI) implies QNL via a $C^{\alpha}$ tree argument, enabling parabolic applications and Fourier bounds for Patterson–Sullivan measures on cusped hyperbolic surfaces. The methods extend to parabolic dynamics through inducing, yielding Fourier decay for Manneville–Pommeau and Lorenz-type maps and, via Li–Pan coding, for Patterson–Sullivan measures; the approach also provides multiscale non-concentration results essential to bounding oscillatory integrals. Overall, the work advances Fourier-analytic methods at the $C^{1+\alpha}$ threshold, constructs explicit non-conjugate-to-linear examples with positive Fourier dimension, and broadens the applicability to parabolic and geometrically finite settings with explicit dependence on tail, UNI, and distortion data.

Abstract

We prove power Fourier decay for equilibrium states of nonlinear $C^{1+α}$ iterated function systems, giving first examples of truly $C^{1+α}$ IFSs with attractors of positive Fourier dimension that are not conjugated to a linear IFS previously out of reach of existing techniques. The key new ingredient is a Measure Non-Linearity (MNL) criterion tailored to $C^{1+α}$ IFSs, coupling oscillations in geometric and symbolic directions and allowing us to prove the necessary multiscale non-linearity condition for Fourier decay via an induction-on-scales scheme. This bypasses the classical obstructions that require either large dimension enabling stationary phase comparisons (Kaufman measures), or smoothness or cohomological conditions on the roof function $τ$ such as $τ\in C^1$ (Dolgopyat's method), $τ\in C^α$ but with smooth underlying dynamics (Tsujii--Zhang strategy), or non-flat $C^α$ conjugacy to a constant (fractal van der Corput lemma). The method also applies to parabolic systems including limit sets of geometrically finite groups, Manneville-Pommeau maps, and Lorenz-type maps.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark
• Theorem 1.5
• Theorem 1.6
• Remark
• Proposition 2.1
• proof : Proof of Proposition \ref{['prop:moments']}