We investigate semiclassical measures for higher dimensional quantum cat maps, which
are standard toy models in quantum chaos. If the matrix quantized has a spectral gap (i.e. a simple
eigenvalue of largest absolute value) and its characteristic polynomial is irreducible over
the rationals, then we show that all semiclassical measures have full support. The spectral gap assumption
makes it possible for us to still use the one-dimensional fractal uncertainty principle in this higher
dimensional situation. When the characteristic polynomial is reducible over rationals, we obtain
a more complicated to state result and compare it with existing counterexamples.
We show lower bounds on mass of eigenfunctions on negatively curved surfaces,
and more generally surfaces with Anosov geodesic flow. We also obtain observability/control for the Schrödinger equation and exponential decay for the damped wave equation.
The proofs partially use the strategy of the previous work in constant curvature, however the case of variable curvature presents substantial challenges due to which a large part of the proof proceeds in a very different way. [Diagram of dependence of sections]
Using the fractal uncertainty principle proved in a previous paper with Bourgain,
we show that every semiclassical defect measure on a compact hyperbolic surface
has support equal to the entire cosphere bundle.
We show that fractal uncertainty principle implies an essential spectral gap
for convex co-compact hyperbolic surfaces. This partially recovers
the result of the paper with Zahl.
In contrast to the latter paper, which relied on complicated microlocal techniques,
our proof uses transfer operators and is relatively short and self-contained.
For a convex co-compact hyperbolic surface with limit set Λ of dimension δ>0, we show that the Fourier dimension of Λ is at least ε for some ε>0 depending only on δ. The proof uses the discretized sum-product theorem in R and the nonlinear nature of the transformations in the group Γ; in fact,
our result would be false for linear Cantor sets. Using fractal uncertainty principle,
we deduce an essential spectral gap of size 1/2-δ+ε.
We show that every Ahlfors–David regular subset of R of positive dimension
satisfies the fractal uncertainty principle with exponent strictly larger than the one coming from the volume bound.
The method of proof is inspired by the works of Dolgopyat, Naud, and Stoyanov in the context
of spectral gaps for transfer operators, and can be crudely summarized as follows:
triangle inequality in a Hilbert space is rarely sharp. As an application we obtain
new spectral gaps for convex co-compact hyperbolic surfaces and open quantum baker's maps.
We show that every convex co-compact hyperbolic surface has an essential spectral gap.
The proof uses the approach developed in the paper with Zahl
and the key new component is the fractal uncertainty principle for δ-regular sets with all δ<1. Previously such gaps were known only under the relaxed pressure
We study open quantum baker's maps, which are models in open quantum chaos having Cantor sets as their trapped sets.
Using the fractal uncertainty approach developed in the paper with Zahl
and the arithmetic structure of Cantor sets, we show that all such systems have a spectral gap,
and obtain quantitative bounds on the size of the gap. We also show
an improved Weyl bound similar to the one here.
Both results are supported by numerical evidence.
We give a Weyl upper bound on the number of scattering resonances on convex co-compact hyperbolic quotients in strips. The exponent in the bound depends on the width of the strip and improves on the standard Weyl upper bound, in particular this exponent is negative until the Patterson–Sullivan gap. The appendix gives numerical evidence for the new Weyl bound.
We obtain an improved spectral gap for convex co-compact hyperbolic surfaces with the dimension of the limit set close to 1/2.
Compared to previous works on spectral gaps which rely on Dolgopyat's method, we decouple the combinatorial difficulties of
the problem from the analytical ones and obtain an explicit formula for the gap in terms of additive energy of the limit set,
which in turn is estimated using its Ahlfors–David regularity.
We use radial estimates (a version for vector bundles)
to investigate what Sobolev regularity is needed for the meromorphic continuation of the Pollicott–Ruelle resolvent for an Anosov flow, in case of an arbitrary
lift of the flow to a vector bundle. The resulting condition is similar
to the one obtained in the paper of
Bonthonneau–Lefeuvre for Hölder–Zygmund regularity.
We study the vanishing order of the Ruelle zeta function at zero for a compact connected oriented
negatively curved 3-manifold. We show that if the metric is a small perturbation of the hyperbolic metric,
then the order of vanishing is equal to 4-b1 where b1 is the first Betti number.
This is in contrast with the hyperbolic case where this order is 4-2b1 and with
the case of surfaces where the previous paper with Zworski showed that the order only depends
on the genus, not on the metric.
Using previous work with Zworski as well
as relatively standard microlocal techniques,
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, giving a relation between dynamics
and topology, was previously known only in constant curvature via the Selberg trace formula.
We define Pollicott–Ruelle resonances for hyperbolic flows on noncompact manifolds (in a setting
closely related to basic sets of Axiom A flows). They arise either as poles of Fourier–Laplace transforms
of correlations or as poles of meromorphic extensions of dynamical zeta functions. Our construction is based
on the microlocal approach of Faure–Sjöstrand
We show that Pollicott–Ruelle resonances of an Anosov flow are the limits of the L2 eigenvalues
of the generator of the flow with an additional damping term. The latter eigenvalues characterize decay of correlations
for a stochastic modification of the flow.
We describe the Pollicott–Ruelle resonances for compact hyperbolic quotients in all dimensions. In dimensions greater than 2,
this description is much more involved than for surfaces and features in particular the spectrum of the Laplacian on trace-free
divergence-free symmetric tensors of all orders.
We prove meromorphic continuation of the Ruelle dynamical zeta function for Anosov flows, recovering
recent results of Giulietti, Liverani, and Pollicott.
We follow the approach of Faure–Sjöstrand; the key new component is the restriction on the wavefront
set of the resolvent, using the methods of Melrose and Vasy, making it possible to take the flat trace.
We give an upper bound on the number of resonances in strips for general
manifolds with Euclidean infinite ends, without any assumptions on the trapped set.
Our bound depends on the volume of the set of trajectories which are trapped for
the Ehrenfest time, and gives a polynomial improvement over the standard bound
Rd-1 when the escape rate is positive. We also prove a wave decay result with
high probability for random initial data.
We obtain a polynomial lower bound on the norm of the scattering resolvent in the lower half-plane under very mild assumptions on trapping, namely existence of a trajectory which is trapped in the past, but not in the future.
The power in the bound, depending on the maximal expansion rate along the trajectory,
gives the smallest number of derivatives lost in an exponential decay of local energy estimate
for the wave equation.
We show that in the presence of certain elliptic trapping, one-sided bounds on the resolvent with a cutoff supported far away into the infinity are at best exponential. This
is in contrast with the known polynomial two-sided bounds.
We show existence of a band of resonances with a Weyl law when the trapped
set is r-normally hyperbolic for large r and the normal expansion
rates are half-pinched. This dynamical setting is stable under perturbations and is applied
the next paper
to Kerr–de Sitter black holes. The key tool is a Fourier integral operator microlocally projecting
onto resonant states in the band. This operator lets us microlocally represent resonance expansions as
Taylor expansions; a variety of methods from microlocal analysis, especially positive commutator estimates, finish the proof.
The r-normal hyperbolicity assumption is explicitly needed in the construction of the projector to ensure smoothness
of its symbol.
We obtain a fractal upper bound on the number of resonances in disks of fixed size centered at
the unitarity axis for a general class of manifolds, including convex co-compact hyperbolic quotients.
The exponent in the bound is related to the dimension of the trapped set.
We use the techniques of Sjöstrand–Zworski
on resonance counting for hyperbolic trapped sets and of
on effective meromorphic continuation of the resolvent for asymptotically hyperbolic
It was proved by Bindel and Zworski in this paper
that for scattering on the half-line by a compactly supported
potential with a constant bump at the end of its support,
the purely imaginary poles of the resolvent
become symmetric with respect to the real axis modulo
errors exponentially small in the semiclassical limit.
Our paper is an extension
of this result to a more general class of potentials,
allowing any positive bump, and also provides a different
explanation of why this phenomenon holds.
Motivated by Kerr and Kerr–de Sitter black holes, we show existence of spectral gaps of optimal size for normally hyperbolic
trapped sets with codimension 1 smooth stable/unstable foliations. The short proofs, based on semiclassical defect measures,
simplify and refine some of the earlier results of Wunsch–Zworski,
Nonnenmacher–Zworski (in our special case), and
We apply the methods and results of a previous paper
to the setting of Kerr–de Sitter black holes, proving a Weyl law for resonances and an asymptotic
decomposition of linear waves for high frequencies. The latter fact, related to resonance expansion,
is also proved for Kerr black holes using the work of Vasy–Zworski.
We establish a Bohr–Sommerfeld type quantization condition for quasi-normal
modes of a slowly rotating Kerr–de Sitter black hole, observing in particular
a Zeeman-like splitting once spherical symmetry is broken. We compute the resulting
pseudopoles numerically [click here for MATLAB codes and data]
and compare them to those
numerically studied by physicists.
Finally, we prove a resonance decomposition of linear waves.
This paper constructs an analogue of the scattering resolvent in the case of
a slowly rotating Kerr–de Sitter black hole. The poles are proven to form
a discrete set; they are the
of black holes that have been
numerically studied by physicists.
Using the recent result by Wunsch and Zworski,
we prove existence of a resonance free strip and exponential energy decay for the wave
equation on a fixed compact set.
Using our previous work, we derive scattering
phase asymptotics for manifolds with asymptotically Euclidean ends with remainders that
are fractal for hyperbolic flows. We next derive asymptotics with a similar remainder
for hyperbolic quotients, and non-sharp lower bounds on the remainder in some cases.
Finally, we investigate the relation of scattering phase asymptotics with counting resonances
near the unitarity axis, in particular with the fractal Weyl law.
For Riemannian manifolds that are either Euclidean or hyperbolic near infinity, and
with trapped set of Liouville measure zero, we
show that plane waves, also known as Eisenstein functions, converge to some semiclassical
measure, if one averages in frequency in an h-sized window and in
the direction of the wave. The speed of convergence is estimated in terms
of the classical escape rate and the Ehrenfest time; in many cases,
this is a power of h (in contrast with quantum ergodicity on compact manifolds,
where the best known result is 1/|log h|). As an application,
we derive a local Weyl law for spectral projectors with a fractal remainder.
We show that on a compact Riemannian manifold with ergodic geodesic flow,
restrictions of the eigenfunctions of the Laplacian to any hypersurface
satisfying a simple geometric condition are equidistributed in phase space.
This work generalizes a paper of
and Steve Zelditch using
the methods of semiclassical analysis, and provides a shorter proof.
We consider a Riemannian surface with cusp ends and show that the Eisenstein functions
in the upper half-plane, away from the real line, converge to a certain canonical
measure. This statement is similar to quantum unique ergodicity(QUE); however,
being away from the real line considerably simplifies the problem. In particular,
no global dynamical properties of the flow are used. As an application,
we prove that the scattering matrix converges to zero in any strip away from the real line.
We study a forcing problem for a 0th order pseudodifferential operator on a surface whose reduced Hamiltonian flow is a Morse–Smale flow without fixed points. In particular we explain how some of the results of a recent paper of Colin de Verdière and Saint-Raymond can be obtained using
We prove certain weighted L-infinity estimates for eigenfunctions
on a strictly convex surface of revolution. Our application lies
in the area of compressed sensing — these estimates
give an improvement on how many random sampling points (chosen with
respect to the measure related to the weight) are enough to recover,
with high probability, a function whose expansion in spherical
harmonics is sparse.
These notes give a self-contained proof of the Stable/Unstable Manifold Theorem (also known
as the Hadamard–Perron Theorem) in hyperbolic dynamics. They also give two examples of hyperbolic systems:
geodesic flows on negatively curved surfaces and dispersing billiards. The proof of the Stable/Unstable Manifold Theorem
starts with a basic model case which however retains the essence of the general case. There are many figures throughout
the text, and it is indended as a (somewhat) gentle introduction to some techniques in hyperbolic dynamics.
An expository article for the proceedings of Journées EDP (Roscoff, June 2017)
describing the results and the proofs of previous works with Bourgain and Jin on fractal uncertainty
principle and its application to control of eigenfunctions.
This is a short introduction to nontrapping estimates in scattering theory.
(1) how one can obtain exponential decay in obstacle scattering
from a nontrapping estimate via the contour deformation argument
(2) how to prove the semiclassical propagation of singularities estimate
in the presence of a complex absorbing potential via Hörmander's
positive commutator method
(3) how propagation of singularities and complex scaling lead to a nontrapping
estimate in the one-dimensional model case.