Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay

Abstract

Let $\Gamma$ be a Zariski dense Kleinian Schottky subgroup of PSL2(C). Let $\Lambda (\Gamma )$ be its limit set, endowed with a Patterson-Sullivan measure $\mu$ supported on $\Lambda (\Gamma )$. We show that the Fourier transform $\hat{\mu }(\xi )$ enjoys polynomial decay as $|\xi |$ 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.

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References in corpus (4)

• Spectral gaps, additive energy, and a fractal uncertainty principle
• Fourier dimension and spectral gaps for hyperbolic surfaces
• Free Groups in Lattices
• Dolgopyat's method and the fractal uncertainty principle

Cited by in corpus (5)

• An introduction to fractal uncertainty principle
• Fourier transform of self-affine measures
• Julia sets of hyperbolic rational maps have positive Fourier dimension
• Discretized Sum-product and Fourier decay in ${\mathbb{R}}^n$
• The fractal uncertainty principle via Dolgopyat's method in higher dimensions