89 citations · 1.1k across the 43 of their papers we have counts for
48 papers
A 1/64 spectral gap for surfaces with
Alex Cohen, Semyon Dyatlov
We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series has an essential spectral gap of size $β=\tfrac…
WKB structure in a scalar model of flat bands
Semyon Dyatlov, Henry Zeng, Maciej Zworski
We consider a family of periodic scalar operators for which one can define flat bands in the sense of Floquet-Bloch theory. One puzzling question originating in recent physics lite…
Semiclassical measures for complex hyperbolic quotients
Jayadev Athreya, Semyon Dyatlov, Nicholas Miller
We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical sys…
Fractal uncertainty in higher dimensions: notes on Cohen's paper
Semyon Dyatlov
In [arXiv:2305.05022], Cohen proved a higher dimensional fractal uncertainty principle for line porous sets. The purpose of this expository note is to provide a different point of…
Mathematics of internal waves in a 2D aquarium
Semyon Dyatlov, Jian Wang, Maciej Zworski
Following theoretical and experimental work of Maas et al we consider a linearized model for internal waves in effectively two dimensional aquaria. We provide a precise description…
Macroscopic limits of chaotic eigenfunctions
Semyon Dyatlov
We give an overview of the interplay between the behavior of high energy eigenfunctions of the Laplacian on a compact Riemannian manifold and the dynamical properties of the geodes…
Semiclassical measures for higher dimensional quantum cat maps
Semyon Dyatlov, Malo Jézéquel
Consider a quantum cat map associated to a matrix , which is a common toy model in quantum chaos. We show that the mass of eigenfunction…
Pollicott-Ruelle resolvent and Sobolev regularity
Semyon Dyatlov
In this note we compute the threshold regularity for meromorphic continuation of the Pollicott--Ruelle resolvent of an Anosov flow as an operator on anisotropic Sobolev spaces, in…
Around quantum ergodicity
Semyon Dyatlov
We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.
The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds
Mihajlo Cekić, Benjamin Delarue, Semyon Dyatlov +1
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold with Betti number , the order of vanishing of the Ruelle zeta function at zero…
Control of eigenfunctions on surfaces of variable curvature
Semyon Dyatlov, Long Jin, Stéphane Nonnenmacher
We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov g…
An introduction to fractal uncertainty principle
Semyon Dyatlov
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 t…
Microlocal analysis of forced waves
Semyon Dyatlov, Maciej Zworski
We use radial estimates for pseudodifferential operators to describe long time evolution of solutions to where is a self-adjoint 0th order pseudodifferent…
Notes on hyperbolic dynamics
Semyon Dyatlov
These expository notes present a proof of the Stable/Unstable Manifold Theorem (also known as the Hadamard--Perron Theorem). They also give examples of hyperbolic dynamics: geodesi…
Dynamical zeta functions for Axiom A flows
Semyon Dyatlov, Colin Guillarmou
We show that the Ruelle zeta function of any smooth Axiom A flow with orientable stable/unstable spaces has a meromorphic continuation to the entire complex plane. The proof uses t…
Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
Semyon Dyatlov
This expository article, written for the proceedings of the Journées EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:…
Fractal uncertainty for transfer operators
Semyon Dyatlov, Maciej Zworski
We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists for which the Selberg zeta function for a convex c…
Fourier dimension and spectral gaps for hyperbolic surfaces
Jean Bourgain, Semyon Dyatlov
We obtain an essential spectral gap for a convex co-compact hyperbolic surface which depends only on the dimension of the limit set. More precisely,…
Semiclassical measures on hyperbolic surfaces have full support
Semyon Dyatlov, Long Jin
We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bun…
Fractal Weyl laws and wave decay for general trapping
Semyon Dyatlov, Jeffrey Galkowski
We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any stron…
Dolgopyat's method and the fractal uncertainty principle
Semyon Dyatlov, Long Jin
We show a fractal uncertainty principle with exponent , , for Ahflors-David regular subsets of of dimension . This improves over the volume bo…
Spectral gaps without the pressure condition
Jean Bourgain, Semyon Dyatlov
For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has…
Resonances for open quantum maps and a fractal uncertainty principle
Semyon Dyatlov, Long Jin
We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions . We show that the size of the spectral gap is…
Ruelle zeta function at zero for surfaces
Semyon Dyatlov, Maciej Zworski
We show that the Ruelle zeta function for a negatively curved oriented surface vanishes at zero to the order given by the absolute value of the Euler characteristic. This result wa…