[Paper Review] Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Jialun Li, Frédéric Naud+1 more|arXiv (Cornell University)|Feb 4, 2019

Mathematical Dynamics and Fractals35 references4 citations

TL;DR

This paper establishes polynomial Fourier decay for Patterson-Sullivan measures on the limit sets of Zariski-dense Kleinian Schottky groups in PSL₂(ℂ), extending Bourgain-Dyatlov's result from the Fuchsian setting. Using exponential sum estimates based on the Bourgain-Gamburd sum-product theorem on ℂ, and proving a non-concentration condition via representation theory and stationary measure regularity, the authors show that the Fourier transform of the measure decays polynomially at infinity, confirming positive Fourier dimension for these fractal sets.

ABSTRACT

Let $Γ$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $Λ(Γ)$ be its limit set, endowed with a Patterson-Sullivan measure $μ$ supported on $Λ(Γ)$. We show that the Fourier transform $\widehatμ(ξ)$ enjoys polynomial decay as $\vert ξ\vert$ goes to infinity. This is a PSL2(C) version of the result of Bourgain-Dyatlov [8], and uses the decay of exponential sums based on Bourgain-Gamburd sum-product estimate on C. These bounds on exponential sums require a delicate non-concentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.

Motivation & Objective

• To establish polynomial Fourier decay for Patterson-Sullivan measures on limit sets of Zariski-dense Kleinian Schottky groups in PSL₂(ℂ).
• To extend the Bourgain-Dyatlov result on Fourier decay in the Fuchsian case to the full PSL₂(ℂ) setting.
• To prove a non-concentration hypothesis for stationary measures on linear groups using representation theory and regularity estimates.
• To demonstrate that such limit sets have positive Fourier dimension, providing deterministic examples of Salem-type sets.

Proposed method

• Uses the Bourgain-Gamburd sum-product estimate on ℂ to bound exponential sums associated with the measure.
• Applies a non-concentration condition on stationary measures of random walks on linear groups, proven via representation theory and regularity estimates.
• Constructs a sequence of positive functions R_n on the limit set that decay uniformly to zero, ensuring the measure is stationary with finite exponential moments.
• Employs the Cartan decomposition to relate operator norms of group elements to geometric radii r_γ, enabling moment estimates.
• Uses a dyadic decomposition of the limit set into cylinders and applies a recursive construction to control the ratio of R_n values.
• Establishes that the group generated by the support of the stationary measure is the full group Γ, using a classical geodesic covering argument.

Experimental results

Research questions

• RQ1 Do limit sets of Zariski-dense Kleinian Schottky groups in PSL₂(ℂ) admit measures with polynomial Fourier decay?
• RQ2 Can the sum-product estimate over ℂ be adapted to control exponential sums for Patterson-Sullivan measures in the non-Fuchsian setting?
• RQ3 What non-concentration condition must be satisfied by stationary measures on linear groups to ensure Fourier decay?
• RQ4 Is the Fourier transform of the Patterson-Sullivan measure on such limit sets uniformly bounded by a negative power of |ξ| as |ξ| → ∞?
• RQ5 Can deterministic, non-trivial fractal sets in ℂ with positive Fourier dimension be constructed via Kleinian group dynamics?

Key findings

• The Fourier transform of the Patterson-Sullivan measure μ on the limit set Λ_Γ satisfies |ŵ(ξ)| = O(|ξ|^{-ε}) for some ε > 0 as |ξ| → ∞.
• The decay rate ε depends only on the measure μ, and the implied constant C depends on the neighborhood 𝒰, the C² norm of φ, and the group Γ.
• The non-concentration hypothesis required for the sum-product argument is verified using representation theory and regularity estimates for stationary measures.
• The exponential moment ∑ν(γ)‖γ‖^ε₁ is finite for small enough ε₁ > 0, ensuring the measure is well-behaved under group action.
• The group Γ_ν generated by the support of the stationary measure ν equals the full group Γ, confirming the measure is fully supported on the dynamics.
• The construction yields a stationary measure ν such that 1 = ∑ν(γ)f_γ on Λ_Γ, confirming the Patterson-Sullivan measure is ν-stationary.