An introduction to fractal uncertainty principle

Dyatlov, Semyon

doi:10.1063/1.5094903

Cited by 27 publications

(41 citation statements)

References 57 publications

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“…is a finite set. The proof is based on the general phenomenon of "fractal uncertainty principle", see [Dya19]. We point out that > 0 can be made explicit, see Jin-Zhang [JZ17] and also Dyatlov-Jin [DJ18].…”

Section: Introductionmentioning

confidence: 91%

Explicit spectral gaps for random covers of Riemann surfaces

Magee

Naud

2020

Publ.math.IHES

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We introduce a permutation model for random degree n covers Xn of a non-elementary convex-cocompact hyperbolic surface X = Γ\H. Let δ be the Hausdorff dimension of the limit set of Γ. We say that a resonance of Xn is new if it is not a resonance of X, and similarly define new eigenvalues of the Laplacian.We prove that for any > 0 and H > 0, with probability tending to 1 as n → ∞, there are no new resonances s = σ + it of Xn with σ ∈ [ 3 4 δ + , δ] and t ∈ [−H, H]. This implies in the case of δ > 1 2 that there is an explicit interval where there are no new eigenvalues of the Laplacian on Xn. By combining these results with a deterministic 'high frequency' resonance-free strip result, we obtain the corollary that there is an η = η(X) such that with probability → 1 as n → ∞, there are no new resonances of Xn in the region { s : Re(s) > δ − η }.

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Section: Introductionmentioning

confidence: 91%

Explicit spectral gaps for random covers of Riemann surfaces

Magee

Naud

2020

Publ.math.IHES

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“…Many versions of the uncertainty principles exist (see [10]), and more recent versions start to take into account the geometry of the time-frequency domains. In particular, in [6], Dyatlov describes the development and applications of a fractal uncertainty principle (FUP) for the separate time-frequency representation, first introduced and developed in [3,7,8]. The relevant localization operator is the standard composition of projections π T Q , where π T and Q project onto the sets T in time and in frequency, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In the context of the FUP, the sets T and take the form of fractal sets. Here fractal sets are defined in terms of the general notion of δregularity (see [6,Definition 2.2. ]), as families of sets T (h), (h) ⊆ [0, 1], dependent on a continuous parameter 0 < h ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

“…An illustrative example featured in [6] is the mid-third Cantor set, where both the time and frequency domain can be regarded as h-neighbourhoods of the Cantor set, say C(h). Notice that the FUP is originally framed such that the parameter h is also encoded in the Fourier transform F h (not unlike the normalization with Planck's constant in quantum mechanics) such that F h f (ω) = 1…”

Section: Introductionmentioning

confidence: 99%

“…In the separate representation, Dyatlov considers the following compositions of projections onto such time and frequency sets, χ Ω(h) F h χ T (h) , where χ E is the characteristic function of a subset E ⊆ R, and F h is a dilated Fourier transform F h f (ω) = √ h −1 F f (ωh −1 ). For signals f ∈ L 2 (R), the FUP then states (see Theorem 2.12 and 2.13 in [2]) that for a fixed 0 < δ < 1 there exists some non-trivial exponent β > max{0, 1/2 − δ} such that…”

Section: Introductionmentioning

confidence: 99%

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Daubechies’ Time–Frequency Localization Operator on Cantor Type Sets I

Knutsen

2020

J Fourier Anal Appl

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We study Daubechies' time-frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time-frequency representation, we define the n-iterate midthird spherically symmetric Cantor set in the joint representation. For the n-iterate Cantor set, precise asymptotic estimates for the operator norm are then derived up to a multiplicative constant.

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Semiclassical Measures for Higher-Dimensional Quantum Cat Maps

Dyatlov

Jézéquel

2023

Ann. Henri Poincaré

Self Cite

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Consider a quantum cat map M associated with a matrix $$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$ A ∈ Sp ( 2 n , Z ) , which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with $$n=1$$ n = 1 , but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.

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An introduction to fractal uncertainty principle

Cited by 27 publications

References 57 publications

Explicit spectral gaps for random covers of Riemann surfaces

Explicit spectral gaps for random covers of Riemann surfaces

Daubechies’ Time–Frequency Localization Operator on Cantor Type Sets I

Semiclassical Measures for Higher-Dimensional Quantum Cat Maps

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