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An introduction to fractal uncertainty principle

Semyon Dyatlov

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.

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.

Paper Structure

This paper contains 17 sections, 26 theorems, 133 equations, 16 figures.

Table of Contents

  1. General results on FUP
  2. Uncertainty principle
  3. Fractal sets
  4. Statement of FUP
  5. Schottky limit sets
  6. FUP with a general phase
  7. Applications of FUP
  8. Control of eigenfunctions
  9. Spectral gaps
  10. FUP for discrete Cantor sets
  11. Proof of discrete FUP
  12. Relation to Fourier decay and additive energy
  13. Fourier decay
  14. Additive energy
  15. FUP in higher dimensions
  16. ...and 2 more sections

Key Result

Proposition 2.10

Fix $0\leq\alpha_{\min}\leq\alpha_{\max}\leq\infty$. 1. Assume that $X$ is $\delta$-regular with constant $C_R$ on scales $\alpha_{\min}$ to $\alpha_{\max}$, and $\delta<1$. Then $X$ is $\nu$-porous on scales $C\alpha_{\min}$ to $\alpha_{\max}$ where $\nu>0$ and $C$ depend only on $\delta,C_R$. 2. A

Figures (16)

  • Figure 1: Left: a model situation to which FUP applies, with the blue/red sets corresponding to localization in position/frequency to a neighborhood of the middle third Cantor set. FUP states that no quantum wavefunction can be localized on both the red and the blue set. Right: a sample of two fractal sets used in applications of FUP to quantum chaos, consisting of trajectories of a hyperbolic dynamical system which do not cross some open set in forward/backward time direction -- see Figure \ref{['f:porosity']} below for details.
  • Figure 2: An example of a Schottky limit set, picturing the sets in \ref{['e:schottky-X']} for $n=1,2,3$. The Schottky data are as follows: $I_1=[-4,-3]$, $I_2=[-2,-1]$, $I_3=[0,1]$, $I_4=[2,3]$, $\gamma_1=\left(7-3-21\right)$, $\gamma_2=\left(3-7-25\right)$. The bottom row is a union of 36 intervals, illustrating the difficulty of plotting a 'nonlinear' fractal set.
  • Figure 3: Heat plots of two high frequency eigenfunctions of the Laplacian on a genus 2 hyperbolic surface (on a fundamental domain inside the Poincaré disk model of the hyperbolic plane), showing equidistribution consistent with the Quantum Unique Ergodicity conjecture. Pictures courtesy of Alexander Strohmaier, produced using the method of Strohmaier--Uski StrohmaierUski.
  • Figure 4: The sets $\Gamma_\pm(T)$ (in black) with the flow direction removed. This figure was produced using numerics for the closely related Arnold cat map $(x,y)\mapsto (2x+y,x+y)$ on the torus $\mathbb R^2/\mathbb Z^2$. The 'hole' $\pi^{-1}(\Omega)$ is the white disk pictured on the leftmost figures. We see that the sets $\Gamma_+(T)$ are smooth in the unstable direction $\mathbb R(2,\sqrt{5}-1)$. Similarly the sets $\Gamma_-(T)$ are smooth in the stable direction $\mathbb R(2,-1-\sqrt 5)$.
  • Figure 5: A tessellation of the hyperbolic plane $\mathbb H^2$ (pictured here in the Poincaré disk model, with $\dot{\mathbb R}$ pictured as the unit circle) by fundamental domains of a convex co-compact group $\Gamma$ constructed using the procedure in §\ref{['s:schottky']} with $r=2$. The quotient $\Gamma\backslash\mathbb H^2$ is a surface with three funnel ends. The fundamental domain defined in \ref{['e:fund-domain']} is shaded. The dashed line is a geodesic on $\mathbb H^2$ whose limiting starting point (denoted by the circle) does not lie in the limit set $\Lambda_\Gamma$ but whose limiting ending point (denoted by the triangle) lies in the limit set. The projection of this geodesic to the quotient is trapped as $t\to\infty$ but not as $t\to-\infty$.
  • ...and 11 more figures

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 48 more