arXiv subjects

Semyon Dyatlov

Publications and source records attributed to Semyon Dyatlov.

At least 37 records · Page 2Linked to original sources

arXiv2017

Spectral gaps without the pressure condition

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 only finitely many zeroes. We make no assumption on the dimension $δ$ of the limit set, in particular we do not require the pressure condition $δ\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $δ$-regular sets with $δ<1$, which may be of independent interest.

math.CA↗

arXiv2017

Fractal Weyl laws and wave decay for general trapping

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 strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time.

math.AP↗

arXiv2017

Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle

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:1705.05019]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.

math.AP↗

arXiv2017

Fourier dimension and spectral gaps for hyperbolic surfaces

We obtain an essential spectral gap for a convex co-compact hyperbolic surface $M=Γ\backslash\mathbb H^2$ which depends only on the dimension $δ$ of the limit set. More precisely, we show that when $δ>0$ there exists $\varepsilon_0=\varepsilon_0(δ)>0$ such that the Selberg zeta function has only finitely many zeroes $s$ with $\Re s>δ-\varepsilon_0$. The proof uses the fractal uncertainty principle approach developed by Dyatlov-Zahl [arXiv:1504.06589]. The key new component is a Fourier decay bound for the Patterson-Sullivan measure, which may be of independent interest. This bound uses the fact that transformations in the group $Γ$ are nonlinear, together with estimates on exponential sums due to Bourgain which follow from the discretized sum-product theorem in $\mathbb R$.

math.CA↗

arXiv2017

Improved fractal Weyl bounds for hyperbolic manifolds

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $δ$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-β,0]$ is $O(R^{m(β,δ)+})$, where the exponent $m(β,δ)=\min(2δ+2β+1-n,δ)$ changes its behavior at $β=(n-1-δ)/2$. In the case $δ<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ λ > δ-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.

math.SP↗

arXiv2017

Ruelle zeta function at zero for surfaces

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 was previously known only in constant curvature.

math.DS↗

arXiv2017

Resonances for open quantum maps and a fractal uncertainty principle

We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions $δ\in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\max(0,{1\over 2}-δ)$ for all values of $δ$, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.

math.SP↗

arXiv2016

Stochastic stability of Pollicott-Ruelle resonances

Pollicott-Ruelle resonances for chaotic flows are the characteristic frequencies of correlations. They are typically defined as eigenvalues of the generator of the flow acting on specially designed functional spaces. We show that these resonances can be computed as viscosity limits of eigenvalues of second order elliptic operators. These eigenvalues are the characteristic frequencies of correlations for a stochastically perturbed flow.

math.DS↗

arXiv2016

Spectral gaps, additive energy, and a fractal uncertainty principle

We obtain an essential spectral gap for $n$-dimensional convex co-compact hyperbolic manifolds with the dimension $δ$ of the limit set close to $(n-1)/2$. The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.

math.SP↗

arXiv2016

Pollicott-Ruelle resonances for open systems

We define Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.

math.DS↗

arXiv2016

Dynamical zeta functions for Anosov flows via microlocal analysis

The purpose of this paper is to give a short microlocal proof of the meromorphic continuation of the Ruelle zeta function for C^\infty Anosov flows. More general results have been recently proved by Giulietti-Liverani-Pollicott [arXiv:1203.0904] but our approach is different and is based on the study of the generator on the flow as a semiclassical differential operator.

math.DS↗

arXiv2015

Lower resolvent bounds and Lyapunov exponents

We prove a new polynomial lower bound on the scattering resolvent. For that, we construct a quasimode localized on a trajectory $γ$ which is trapped in the past, but not in the future. The power in the bound is expressed in terms of the maximal Lyapunov exponent on $γ$, and gives the minimal number of derivatives lost in exponential decay of solutions to the wave equation.

math.AP↗

arXiv2015

Spectral gaps for normally hyperbolic trapping

We establish a resonance free strip for codimension 2 symplectic normally hyperbolic trapped sets with smooth incoming/outgoing tails. An important application is wave decay on Kerr and Kerr-de Sitter black holes. We recover the optimal size of the strip and give an $o(h^{-2})$ resolvent bound there. We next show existence of deeper resonance free strips under the $r$-normal hyperbolicity assumption and a pinching condition. We also give a lower bound on the ne-sided cutoff resolvent on the real line.

math.AP↗

arXiv2015

Power spectrum of the geodesic flow on hyperbolic manifolds

We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.

math.DS↗

arXiv2014

The sectional curvature remains positive when taking quotients by certain nonfree actions

We study some cases when the sectional curvature remains positive under the taking of quotients by certain nonfree isometric actions of Lie groups. We consider the actions of the groups $S^1$ and $S^3$ such that the quotient space can be endowed with a smooth structure using the fibrations $S^3/S^1{\simeq}S^2$ and $S^7/S^3\simeq S^4$. We prove that the quotient space carries a metric of positive sectional curvature, provided that the original metric has positive sectional curvature on all 2-planes orthogonal to the orbits of the action.

math.DG↗

arXiv2014

Resonance projectors and asymptotics for r-normally hyperbolic trapped sets

We prove an asymptotic formula for the number of scattering resonances in a strip near the real axis when the trapped set is r-normally hyperbolic with r large and a pinching condition on the normal expansion rates holds. Our dynamical assumptions are stable under smooth perturbations and motivated by the setting of black holes. The key tool is a Fourier integral operator which microlocally projects onto the resonant states in the strip. In addition to Weyl law, this operator provides new information about microlocal concentration of resonant states.

math.AP↗

arXiv2014

Resonances and lower resolvent bounds

We show how the presence of resonances close to the real axis implies exponential lower bounds on the norm of the cut-off resolvent on the real axis.

math.AP↗

arXiv2013

Trapping of waves and null geodesics for rotating black holes

We present dynamical properties of linear waves and null geodesics valid for Kerr and Kerr-de Sitter black holes and their stationary perturbations. The two are intimately linked by the geometric optics approximation. For the nullgeodesic flow the key property is the r-normal hyperbolicity of the trapped set and for linear waves it is the distribution of quasi-normal modes: the exact quantization conditions do not hold for perturbations but the bounds on decay rates and the statistics of frequencies are still valid.

gr-qc↗