An introduction to fractal uncertainty principle

Abstract

Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex co-compact hyperbolic surfaces.

arXiv comment45+eps pages, 16 figures. To appear in Journal of Mathematical Physics

References in corpus (21)

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• Spectral gaps without the pressure condition
• Spectral problems in open quantum chaos
• Control of eigenfunctions on surfaces of variable curvature
• Experimental Observation of the Spectral Gap in Microwave n-Disk Systems
• Spectral gaps, additive energy, and a fractal uncertainty principle
• Semiclassical measures on hyperbolic surfaces have full support
• Fourier dimension and spectral gaps for hyperbolic surfaces
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• Resonances for open quantum maps and a fractal uncertainty principle
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• Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay
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• Dolgopyat's method and the fractal uncertainty principle
• A higher dimensional Bourgain-Dyatlov fractal uncertainty principle
• Fractal uncertainty for transfer operators
• Notes on hyperbolic dynamics
• Fourier decay in nonlinear dynamics
• Damped wave equations on compact hyperbolic surfaces
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• Fractal Uncertainty Principle with Explicit Exponent

Cited by in corpus (16)

• Control of eigenfunctions on surfaces of variable curvature
• On pointwise decay of waves
• Semiclassical measures for higher dimensional quantum cat maps
• Polynomial Fourier decay for fractal measures and their pushforwards
• Resonance states of the three-disk scattering system
• Semiclassical Limit of Resonance States in Chaotic Scattering
• Additive energy of regular measures in one and higher dimensions, and the fractal uncertainty principle
• Spectral gap for obstacle scattering in dimension 2
• Fractal uncertainty for discrete 2D Cantor sets
• Inverse theorems for discretized sums and $L^q$ norms of convolutions in ${\mathbb{R}}^d$
• Macroscopic limits of chaotic eigenfunctions
• Quantum Distributions for the Electromagnetic Field
• The fractal uncertainty principle via Dolgopyat's method in higher dimensions
• Fractal uncertainty principle for discrete Cantor sets with random alphabets
• Semiclassical behaviour of quantum eigenstates
• Daubechies' Time-Frequency Localization Operator on Cantor Type Sets