arXiv subjects
Publications and source records attributed to Semyon Dyatlov.
We show that any convex co-compact hyperbolic surface with exponent of convergence of Poincaré series $δ\in (\frac25,\frac{14}{27})$ has an essential spectral gap of size $β=\tfrac78(\tfrac12-δ)+\tfrac{1}{32}δ-ε$ for any $ε>0$. In particular, for $δ=\frac12$ this becomes $β=\tfrac{1}{64}-ε$. We show existence of the gap by proving a new Fractal Uncertainty Principle, using a two-ends Furstenberg theorem of O'Regan-Wu-Yi [arXiv:2607.08461].
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 literature is a quantisation rule for the values of parameters at which these flat bands occur. We present a general theorem about the structure of solutions to the corresponding equation and a heuristic argument explaining their WKB structure in a specific case. That structure also explains the quantisation condition - both the WKB structure and that rule are confirmed by numerical experiments. Finally, we consider a simplified model in which separation of variables allows the use of complex WKB methods.
We study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel [arXiv:2108.10463] in the toy model of quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.
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 of singular profiles appearing in long time wave evolution and associate them to classical attractors. That is done by microlocal analysis of the spectral Poincaré problem, leading in particular to a limiting absorption principle. Some aspects of the paper (for instance Section 6) can be considered as a natural microlocal continuation of the work of John on the Dirichlet problem for hyperbolic equations in two dimensions.
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 view on some parts of Cohen's proof, particularly suited to readers familiar with the theory of distributions. It is meant to be complementary to Cohen's paper.
Consider a quantum cat map $M$ associated to a matrix $A\in\mathop{\mathrm{Sp}}(2n,\mathbb Z)$, 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 [arXiv:1705.05019], [arXiv:1906.08923] on negatively curved surfaces and [arXiv:2103.06633] on quantum cat maps with $n=1$, 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.
We show that for a generic conformal metric perturbation of a compact hyperbolic 3-manifold $Σ$ with Betti number $b_1$, the order of vanishing of the Ruelle zeta function at zero equals $4-b_1$, while in the hyperbolic case it is equal to $4-2b_1$. This is in contrast to the 2-dimensional case where the order of vanishing is a topological invariant. The proof uses the microlocal approach to dynamical zeta functions, giving a geometric description of generalized Pollicott-Ruelle resonant differential forms at 0 in the hyperbolic case and using first variation for the perturbation. To show that the first variation is generically nonzero we introduce a new identity relating pushforwards of products of resonant and coresonant 2-forms on the sphere bundle $SΣ$ with harmonic 1-forms on $Σ$.
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 the setting of lifts to general vector bundles. These thresholds are related to the Sobolev regularity needed for the decay of correlations.
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 geodesic flow on that manifold. This includes the Quantum Ergodicity theorem, the Quantum Unique Ergodicity conjecture, entropy bounds, and uniform lower bounds on mass of eigenfunctions. The above results belong to the domain of quantum chaos and use microlocal analysis, which is a theory behind the classical/quantum, or particle/wave, correspondence in physics. We also discuss the toy model of quantum cat maps and the challenges it poses for Quantum Unique Ergodicity.
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.
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 geodesic flows. This implies controllability for the Schrödinger equation by any nonempty open set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works [arXiv:1705.05019], [arXiv:1712.02692], which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of [arXiv:1705.05019], [arXiv:1712.02692] and rely on the fractal uncertainty principle of [arXiv:1612.09040]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used in [arXiv:1504.06589]. Instead, our argument uses Egorov's Theorem up to local Ehrenfest time and the hyperbolic parametrix of [arXiv:0706.3242], together with the $C^{1+}$ regularity of the stable/unstable foliations.
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.
We use radial estimates for pseudodifferential operators to describe long time evolution of solutions to $ i u_t - P u = f $ where $ P $ is a self-adjoint 0th order pseudodifferential operator satisfying hyperbolic dynamical assumptions and where $ f $ is smooth. This is motivated by recent results of Colin de Verdière and Saint-Raymond [arXiv:1801.05582] concerning a microlocal model of internal waves in stratified fluids.
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 bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in [arXiv:1504.06589] and proved for porous sets in [arXiv:1612.09040].
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: geodesic flows on surfaces of negative curvature and dispersing billiards.
We show directly that the fractal uncertainty principle of Bourgain-Dyatlov [arXiv:1612.09040] implies that there exists $ σ> 0 $ for which the Selberg zeta function for a convex co-compact hyperbolic surface has only finitely many zeros with $ \Re s \geq \frac12 - σ$. That eliminates advanced microlocal techniques of Dyatlov-Zahl [arXiv:1504.06589] though we stress that these techniques are still needed for resolvent bounds and for possible generalizations to the case of non-constant curvature.
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 the meromorphic continuation result of [arXiv:1410.5516] together with the work of Conley-Easton and [arXiv:1711.10059] which imply that every basic hyperbolic set can be put into the framework of [arXiv:1410.5516].
We show a fractal uncertainty principle with exponent $1/2-δ+ε$, $ε>0$, for Ahflors-David regular subsets of $\mathbb R$ of dimension $δ\in (0,1)$. This improves over the volume bound $1/2-δ$, and $ε$ is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Here the group invariance of the set is replaced by its fractal structure. As an application, we quantify the result of Naud on spectral gaps for convex co-compact hyperbolic surfaces and obtain a new spectral gap for open quantum baker maps.