DYNAMICAL SPECTRAL RIGIDITY AMONG
Z2 -SYMMETRIC STRICTLY CONVEX DOMAINS
CLOSE TO A CIRCLE
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Abstract. We show that any sufficiently (finitely) smooth Z2 symmetric strictly convex domain sufficiently close to a circle is
dynamically spectrally rigid, i.e. any deformation among domains
in the same class which preserves the length of all periodic orbits
must necessarily be an isometric deformation.
This is a preliminary version
1. Introduction
In this paper we study questions motivated by the famous question
of M. Kac [11]: “Can one hear the shape of a drum?”. Formally, given
a domain Ω ⊂ R2 , consider the spectrum of the Laplace operator in Ω
with (e.g.) Dirichlet boundary condition1, denote by ∆(Ω), i.e. the set
of λ so that
∆u + λ2 u = 0,
u = 0 on ∂Ω.
“Does the Laplace spectrum determine a domain?” There is a number
of counterexamples to this question (see e.g. [8, 21, 22]). Notably
Sunada [21] constructed isospectral sets of arbitrary large cardinality.
However, the domains considered in such examples are neither smooth
nor convex.
For planar domains with a C ∞ smooth boundary Osgood-PhillipsSarnak [13, 14, 15] showed that an isospectral set of planar domains is
compact in C ∞ topology. Sarnak [20] conjectured that an isospectral
set consists of isolated points, namely, any planar domain Ω with C ∞
smooth boundary cannot be perturbed to an isospectral domain Ω0
unless perturbation is an isometry.
In the affirmative direction Hezari–Zelditch [10] proved that given
an ellipse E, any one-parameter C ∞ -deformation Ωτ which preserves
1
From the physical point of view, the Dirichlet eigenvalues λ correspond to
the eigenfrequencies of a vibrating membrane of shape Ω which is fixed along its
boundary.
1
2
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
the Laplace spectrum (with respect to either Dirichlet or Neumann
boundary conditions) and the Z2 × Z2 symmetry group of the ellipse
has to be flat (i.e., all derivatives have to vanish for τ = 0). Popov–
Topalov [19] extended these results (see also [18]). Further historical
remarks on the inverse spectral problem can also be found in [10].
1.1. The length spectrum and relation with the Laplace spectrum. An object closely related to the Laplace spectrum is so-called
length spectrum. The Length Spectrum of a domain Ω is the set:
L(Ω) = N {length of closed geodesic of Ω} ∪ N {l∂Ω },
where l∂Ω denotes the length of the boundary ∂Ω and N = {1, 2, · · · }.
There is a remarkable relation between the Laplace spectrum and
the length spectrum for convex domains.
eq:laplace-length
Andersson–Melrose (see [1, Theorem (0.5)], which substantially generalizes some earlier result by [3, 7]) proved that, for strictly convex
C ∞ domains, the following relation between the wave trace and the
length spectrum holds:


X
(1.1)
sing supp t 7→
exp(iλj t) ⊂ ±L(Ω) ∪ {0}.
λj ∈∆(Ω)
Generically the above inclusion becomes an equality and the Laplace
spectrum determines the length spectrum (see Remark 2.9).
1.2. Inverse Dynamical Problem. The length spectrum of Ω is
clearly invariant under isometries: it is then natural to ask the following: Can Ω be determined by L(Ω) (modulo isometries)?
The above question turns out to be quite difficult to answer; there
are counterexamples as soon as one drops the convexity assumptions
(see below), but in case of convex sufficiently smooth domain it is an
open problem. We present results on a deformational version of this
question which we call dynamical spectral rigidity.
1.3. Dynamical spectral rigidity. We call a C 1 smooth one-parameter family (Ωτ )τ ∈U dynamically isospectral if L(Ωτ ) = L(Ω0 ) for any
τ ∈ U . A domain is called dynamical spectrally rigid if any dynamically
isospectral family (Ωτ )τ ∈U is a family of isometries.
Consider the space S r of strictly convex domains with axial symmetry whose boundary is C r+1 and r is sufficiently large.
DYNAMICAL SPECTRAL RIGIDITY
3
We prove that any domain Ω ∈ S r with nearly circular boundary is
dynamical spectrally rigid.2
1.4. Local rigidity. Local rigidity problems, i.e. isospectral deformations, were considered in some early works on variations of the spectral functions and wave invariants. It is worth to mention the work
of Guillemin-Kazhdan [9]. They introduce the following definition of
(deformation) spectral rigidity.
Let (M, g) be a compact boundaryless Riemannian manifold. A family (gτ )τ ∈U of Riemannian metrics on M smoothly depending on the
parameter τ ∈ [−ε, ε] is called the deformation of the metric g if g0 = g.
A deformation is called trivial if there exists a one-parameter family of
diffeomorphisms ϕτ : M → M such that ϕ0 = Id, and gτ = (ϕτ )∗ g0 .
Given a deformation gτ (|τ | ≤ ε), let ∆τ : C ∞ (M ) → C ∞ (M ) be the
Laplace-Beltrami operator corresponding to the metric gτ . The deformation is called isospectral if, for all |τ | ≤ ε spectra of the operators
∆τ and ∆0 coincide (counting multiplicities). A Riemannian manifold
(M, g) is called deformationally spectrally rigid if it does not admit
non-trivial isospectral deformations.
They show that a negatively curved surface can’t be isospectrally
deformed among negatively curved surfaces. It is shown that if γ a
closed geodesic of a metric g that is non-degenerate and of mulitiplicity one in the length spectrum, then for any deformation gt of g, γ
deforms smoothly as a closed
R geodesic γt of gt , then the first variation
of the length Lγ is given by γ ġds. For a negative curved surface, there
does not exist a non-trivial
symmetric tensor ġ satisfying the isospecR
tral condition that γ ġds = 0. This result for compact manifolds of
negative curvature was proven in [5]
A global version of the [9] result is to Croke [4] and Otal [16]. Namely,
a properly defined Marked Length spectrum determines a surface of
negative curvature up to an isometry.
Our result is an analogue of [9] and [5] for Z2 symmetric billiard
table close to the disk. (A natural analogue of negative curved surfaces
is hyperbolic billiard table).
Another example of deformational spectral rigidity appears in De la
Llave, Marco and Moriyón [6]. It is shown that there are no non-trivial
deformations of exact symplectic mappings Bτ , τ ∈ [0, 1], leaving the
length spectrum fixed when Bs are Anosov’s mappings on a symplectic
manifold. One of the reasons for symplectic rigidity in [6] is that all
periodic points of Bs are hyperbolic and form a dense set.
2A
more precise formulation will be provided in the next section.
4
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Now we turn to a rigorous statement of our results and their proof.
2. Statement of results
We denote by Dr the set of strictly convex planar domains Ω whose
boundary is C r+1 smooth3 and by S r ⊂ Dr the subset of domains with
axial symmetry.
Definition 2.1. A closed geodesic in Ω is a (not necessarily convex)
polygon inscribed in ∂Ω so that at each vertex, the angles formed by
each of the two sides joining at the vertex with the tangent line to ∂Ω
are equal. The perimeter of the polygon is called the length of the
geodesic.
Definition 2.2. The Length Spectrum4 of a domain Ω is the set:
L(Ω) = N {length of closed geodesic of Ω} ∪ N {l∂Ω },
where l∂Ω is the length of the boundary ∂Ω.
Obviously, the length spectrum of Ω is invariant under isometries: it
is then natural to ask the following
Question (Inverse Dynamical Problem). Can Ω ∈ Dr be determined
by L(Ω) (modulo isometries)?
Our results on a deformational version of this question which we call
dynamical spectral rigidity. In order to state this condition we need
some preparation.
Definition 2.3. We say that (Ωτ )|τ |≤ε is a C 1 one-parameter family
of domains in S r if Ωτ ∈ S r for any |τ | ≤ ε and there exists
γ(τ, s) : U × T1 → R2
so that γ(τ, s) is continuously differentiable in τ with bounded derivativeand, for any τ ∈ U , the map γ(τ, ·) is a C r+1 diffeomorphism of T1
onto ∂Ωτ .
Remark 2.4. Notice that for a given family (Ωτ )|τ |≤ε , the choice of γ is
quite arbitrary: given any family of C r+1 circle diffeomorphisms s(τ, σ)
we can find another parametrization of the same family of domains by
choosing γ̃(τ, σ) = γ(τ, s(τ, σ)).
3
We denote the superscript with r, because the associated billiard map is C r
(see the next section).
4 It possible to give an alternative definition of the length spectrum:
L(Ω) = N {length of closed geodesic of Ω}
DYNAMICAL SPECTRAL RIGIDITY
5
Definition 2.5. A family (Ωτ )|τ |≤ε is said to be isometric (or trivial) if there exists a family (Iτ )τ ∈U of isometries Iτ : R2 → R2 (i.e.
composition of a rotation and a translation) so that Ωτ = Iτ Ω0 .
A question related to the inverse dynamical problem is whether or
not, given a domain Ω, there exist so called isospectral deformations of
Ω, which we now define.
Definition 2.6. A family (Ωτ )|τ |≤ε is dynamically isospectral (or lengthisospectral) if L(Ωτ ) = L(Ω0 ) for any τ ∈ U .
Definition 2.7. A domain Ω0 ∈ Dr (resp. Ω0 ∈ S r ) is said to be
dynamically spectrally rigid (resp. S r -dynamically spectrally rigid) if
any dynamically isospectral family of deformations (Ωτ )τ ∈U ⊂ Dr (resp.
(Ωτ )τ ∈U ⊂ S r ) is an isometric family.
Recall that a closed curve is a circle if and only if its radius of curvature is constant.
Definition 2.8. A domain Ω ∈ Dr of perimeter 1 is said to be δ-close
to a circle if
k2πρ(s) − 1kC r−1 ≤ δ.
Recall that if Ω ∈ Dr , ∂Ω is C r+1 , thus its curvature is C r−1 . A domain
Ω of arbitrary perimeter is said to be δ-close to a circle if its rescaling
of perimeter 1 is δ-close to a circle.
We are now able to state the main result of this paper.
Main Theorem. Let r be sufficiently large and Ω ∈ S r , whose boundary ∂Ω is sufficiently close to a circle, then Ω is S r -dynamically spectrally rigid.
rm:spectra-equality
Remark 2.9. It turns out that the Laplace spectrum generically determines the length spectrum, namely, in (1.1) we have an equality
(see [17]). Generic conditions are as follows:
(1) no two distinct orbits have the same length
(2) the Poincaré map of any periodic orbit of the associated billiard
map (see (3.1)) is non-degenerate.
It is convenient to restate the result as follows:
Definition 2.10. A domain Ω ∈ S r is said to be normalized if the
marked point of ∂Ω is at the origin of R2 , its symmetry axis coincides
with the x-axis and Ω lies in the right half plane. A family (Ωτ )τ ∈U is
said to be normalized if Ωτ is normalized for any τ ∈ U .
6
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Given a family (Ωτ )τ ∈U , we can always construct a normalized family
(Ω̃τ )τ ∈U by isometries as follows:
• translate the domain so that the marked point of ∂Ωt is at the
origin of R2 .
• rotate the domain around the origin so that the symmetry axis
is the x-axis and the domain lies in the right half plane.
By smoothness (in τ ) of (Ωτ )τ ∈U , we gather that (Ω̃τ )τ ∈U can be chosen
to be as smooth as the original family (Ωτ )τ ∈U . We call (Ω̃τ )τ ∈U the
normalization of the family (Ωτ )τ ∈U . Observe that, since Ω̃τ is obtained
from Ωτ via an isometry, we have L(Ωτ ) = L(Ω̃τ ); in particular, if
(Ωτ )τ ∈U is a dynamically isospectral family, so is (Ω̃τ )τ ∈U .
Our Main Theorem can then be restated as follows:
t_main’’
Theorem 2.11. Let r be sufficiently large and (Ωτ )τ ∈U be a normalized
C 1 -family of domains in S r sufficiently C 6 -close to a circle. If L(Ωτ ) =
L(Ω0 ) for all τ ∈ U , then Ωτ = Ω0 .
From now on we will assume that (Ωτ )τ ∈U is a normalized C 1 -family
of S r -domains so that Ω0 has perimeter 1.
Elementary ODE considerations imply that there exists a parametrization γ : U × T1 → R of (Ωτ )τ ∈U which satisfies the following properties:
(1) γ(0, s) is the arc-length parametrization of ∂Ω0
(2) for any τ ∈ U , the point γ(τ, 0) is the marked point of ∂Ωτ ,
that is, the origin of R2 .
(3) for any τ ∈ U and s ∈ T1 , the vector ∂τ γ(τ, s) is orthogonal to
∂Ωτ at γ(τ, s).
We call this parametrization the Canonical parametrization of the family (Ωτ )τ ∈U . Then we define the orthogonal deformation associated to
(Ωτ )τ ∈U to be the function n(τ, s) which satisfies ∂τ γ(τ, s) = n(τ, s)N (τ, s),
where N (τ, s) is the outgoing unit normal vector to ∂Ωτ at γ(τ, s).
Remark 2.12. Observe that if (Ωτ )τ ∈U ⊂ S r is a normalized family, it is generated by orthogonal deformations n(τ, s) that satisfy the
following properties: for any τ ∈ U :
(1) n(τ, ·) is an even function, i.e. n(τ, s) = n(τ, −s);
(2) n(τ, 0) = 0.
We can thus further restate our Main Theorem as
t_main’
Theorem 2.13. Let r be sufficiently large and (Ωτ )τ ∈U be a normalized
C 1 -family of domains in S r sufficiently close to a circle and n the
associated normal perturbation. If L(Ωτ ) = L(Ω0 ) for all τ ∈ U , then
n = 0.
DYNAMICAL SPECTRAL RIGIDITY
7
3. Billiard dynamics of axially symmetric domains
def:bill-map
Let Ω ∈ S r be a domain with an axis of symmetry; for definiteness
we fix the length of its boundary to be equal to 1 and the counterclockwise orientation on ∂Ω to be the positive orientation. We mark
a point P0 ∈ ∂Ω to be one of the two points that lie on the symmetry axis of Ω; we will consider the point P0 to be the origin of all
our parametrizations of ∂Ω. In particular, since Ω has perimeter 1,
the other intersection of ∂Ω with the symmetry axis corresponds to
the arc-length parameter 1/2. We consider the billiard dynamics on
Ω, which is described as follows: a point particle travels with constant
velocity in the interior of Ω; when the particle hits ∂Ω, it is reflected according to the law of optical reflection: angle of incidence equals angle
of reflection. Closed geodesics of Ω are, therefore, in 1-to-1 correspondence to periodic trajectories of the billiard dynamics. It is customary
to study the dynamics by passing to a discrete-time version of it, i.e.
to a map on the canonical Poincaré section M = ∂Ω × [−1, 1]. The
first variable (which is usually parametrized in arc-length, denoted with
s ∈ T1 = R/Z) indicates the point at which the particle has collided
with ∂Ω and the second coordinate Φ equals cos(ϕ), where ϕ is the
angle that the outgoing trajectory forms with the positively oriented
tangent to ∂Ω. The billiard ball map on M is then defined as
f : ∂Ω × [−1, 1] → ∂Ω × [−1, 1]
(3.1)
(s, Φ) 7→ (s0 , Φ0 )
where s0 is the coordinate of the point at which the trajectory emanating from s with angle ϕ collides once again with ∂Ω and Φ0 = cos ϕ0 ,
where ϕ0 is the angle of incidence of the new collision with the positively oriented tangent to ∂Ω at s0 . The map f is an exact twist
diffeomorphism which preserves the area form ds ∧ dΦ. Let us denote
by
L(s, s0 ) = kγ(s) − γ(s0 )k
the Euclidean distance between two points on ∂Ω. Then L is a generating function of the billiard ball map, i.e.
(
∂L
(s, s0 ) = −Φ
∂s
∂L
(s, s0 ) = Φ0 .
∂s0
Having defined the billiard map f we proceed to prove the following
l_spectrumNullSet
Lemma 3.1. The length spectrum of Ω has zero Lebesgue measure.
Proof. Recall that by Sard’s Lemma the set of critical values of a
real valued C r -function defined on an n-dimensional manifold has zero
8
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Lebesgue measure provided that r ≥ n. For any q, let us define the
function
Lq (s, Φ) = L(s1 , s0 ) + L(s2 , s1 ) + · · · + L(s0 , sq−1 )
where (sk , Φk ) = f k (s, Φ). Periodic orbits of period q of the billiard
map correspond to critical points of Lq . Indeed, the law angle of reflection at sk equals to the angle of incidence implies that partial derivative
of Lq with respect to sk equals zero. Therefore, their lengths correspond
to critical values of Lq . Since the billiard map f is C r with r > 2, we
conclude that the set of lengths of periodic orbits of period q has zero
Lebesgue measure; by taking the (countable) union over q we obtain
L(Ω) and we conclude that L(Ω) has zero Lebesgue measure.
Consider a periodic orbit of period q: let p ∈ Z be its winding number
(i.e. the number of times that the associated polygon wraps around
the boundary ∂Ω); then the rotation number of the orbit is defined
as the ratio p/q. The following lemma is a simple consequence of the
fact that Ω has axial symmetry and of the normalization that we have
chosen.
orbit
_variationalProblems
var1
Lemma 3.2. For any q > 1, there exists a periodic orbit, which we
denote with S q (Ω) and call marked symmetric maximal periodic orbit,
of rotation number 1/q and maximal length passing through the origin
s = 0 in ∂Ω.
Proof. We distinguish the cases of even and odd period.
Case 1: q = 2k is even. We claim there exists a q-periodic orbit
passing through s = 0 and s = 1/2. Indeed, let us fix s0 = 0 and
sk = 1/2, and consider the problem of maximizing the function
(3.2a)
q
L (ξ) := 2
k−1
X
L(si , si+1 )
i=0
on the compact set 0 = s0 ≤ s1 ≤ · · · ≤ sk−1 ≤ sk = 1/2, where
ξ = (s1 , · · · , sk−1 ). By the triangle inequality and strict convexity, the
maximum is attained at a critical point (s̄1 , . . . , s̄k−1 ) so that s0 < s1 <
· · · < sk−1 < sk . If we fix conventionally s̄0 = 0, s̄k = 1/2. we have
∂1 L(s̄i , s̄i+1 ) = −∂2 L(s̄i−1 , s̄i ),
i = 1, . . . , k − 1.
Completing s̄2k−i = −s̄i , i = 1, . . . , k − 1, we obtain a periodic orbit of
period 2k = q, which is of maximal length among symmetric orbits.
Case 2: q = 2k + 1 is odd. We claim there exists a periodic orbit
passing through γ(s = 0) and so that the segment γ(sk )γ(sk+1 ) is
DYNAMICAL SPECTRAL RIGIDITY
9
perpendicular to the symmetry axis. Indeed, let us fix s0 = 0 and
consider the problem of maximizing the function
var2
(3.2b)
q
L (ξ) :=
k−1
X
2L(si , si+1 ) + L(sk , −sk )
i=0
on the compact set 0 = s0 ≤ s1 ≤ · · · ≤ sk ≤ 1/2, where ξ =
(s1 , · · · , sk ). Once again by the triangle inequality and strict convexity, the maximum is attained at a critical point (s̄1 , . . . , s̄k ) so that
0 < s̄1 < · · · < s̄k−1 < 1/2. Moreover, if we fix conventionally s̄0 = 0,
we have
0 = ∂1 L(s̄i , s̄i+1 ) + ∂2 L(s̄i−1 , s̄i ), i = 1, . . . , k − 1
1
1
0 = ∂2 L(s̄k−1 , s̄k ) + ∂1 L(s̄k , −s̄k ) − ∂2 L(s̄k , −s̄k )
2
2
= ∂2 L(s̄k−1 , s̄k ) + ∂1 L(s̄k , −s̄k )
Completing s̄2k+1−i = −s̄i , i = 1, . . . k − 1, we obtain a periodic orbit
of period 2k + 1 = q which is of maximal length amongst all symmetric
orbits.
r_fixOrbits
Remark 3.3. In the previous lemma we found symmetric periodic orbits as extremal points of a suitably defined variational problem. In
general such orbits are not uniquely determined5. Even if it is possible
to show that for sufficiently small perturbations of circles, such orbits
are uniformly non-degenerate (i.e. by the implicit function theorem they
all persist for sufficiently small perturbations of Ω), we will not need
non-degeneracy of such orbits to conclude our argument . However,
for any q and τ ∈ U , we will fix once and for all a marked symmetric
maximal periodic orbit which we denote with S q (Ωτ ).
Denote γτ (s) = γ(τ, s) and Lτ (s, s0 ) = kγτ (s)−γτ (s0 )k. For q ≥ 2, let
Lq (τ ; ξ) denote the function defined in (3.2) corresponding to Ωτ . Let
∆q (τ ) denote the length of the marked maximal symmetric periodic
orbit of rotation number 1/q for the domain Ωτ , that is
∆q (τ ) = max Lq (τ ; ξ)
ξ
Lemma 3.4. For any q ≥ 2, the function ∆q (τ ) : U → R is a locally
Lipschitz function.
Proof. Let Ū 0 ⊂ U be any compact subset, and define
Kq := max max k∂τ Lq (τ ; ξ)k .
τ ∈Ū 0
5
ξ
It is easy to construct domains with 1-parameter families of such orbits.
10
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Let us fix τ ∈ Ū 0 , and let ξ¯ realize the maximum of Lq (τ ; ξ), that is
¯ = ∆q (τ ). Then Lq (τ 0 ; ξ)
¯ ≤ ∆q (τ 0 ), for any τ 0 ∈ Ū 0 , hence
Lq (τ, ξ)
¯ − Lq (τ 0 ; ξ)
¯ ≤ Kq |τ − τ 0 |
∆q (τ ) − ∆q (τ 0 ) ≤ Lq (τ ; ξ)
Exchanging τ and τ 0 in the above inequality, we can thus conclude that:
|∆q (τ ) − ∆q (τ 0 )| ≤ Kq |τ − τ 0 |.
Given a continuous function f : U ⊂ R → R, one can define its upper
(resp. lower) differential D+ f (τ ) (resp. D− f (τ )), which is characterized
as follows: p ∈ D+ f (τ ) (resp. p ∈ D− f (τ )) if and only if there exists a
function ψ ∈ C 1 (U, R), such that ψ 0 (τ ) = p and ψ ≥ f (resp. ψ ≤ f ) in
a neighborhood of τ , with equality at τ . Both D+ f (τ ) and D− f (τ ) are
convex subset of R; they are both non empty if and only if D+ f (τ ) =
D− f (τ ) = {f 0 (τ )}.
upp-dif
Lemma 3.5. If ξ¯ is a point realizing the maximum ∆q (τ ) = maxξ Lq (τ ; ξ),
¯ ∈ D− ∆q (τ ).
then we have ∂τ Lq (τ ; ξ)
Proof. Since ∆q (·) is the maximum of Lq (·, ξ), if ξ¯ realizes the maximum
¯ ≤ ∆q (τ 0 ) for any τ 0 ∈ U . Noticing that Lq (·; ξ)
¯ is
at τ we have Lq (τ 0 , ξ)
a C 1 function, by the characterization of lower differential, we conclude
¯ ∈ D− ∆q (τ ).
that p = ∂τ Lq (τ ; ξ)
Remark 3.6. Indeed, we could show that ∆q (t) is a semi-convex function and
¯ q (t; ξ)
¯ = ∆q (t)}
D− ∆q (t) = co{∂t Lq (t; ξ)|L
l_isospectral
Lemma 3.7. If (Ωτ )τ ∈U is an isospectral family, then for any q ≥ 2
and τ ∈ U we have ∆q (τ ) = ∆q (0).
Proof. Assume that for some τ and q ≥ 2 we have ∆q (τ ) 6= ∆q (0); then
since ∆q (τ ) is continuous, ∆q ([0, τ ]) contains an open set. By definition,
for any τ , ∆q (τ ) ∈ L(Ωτ ), but since the family is isospectral, we gather
that ∆q (τ ) ∈ L(Ω0 ), hence ∆q ([0, τ ]) ⊂ L(Ω0 ). Hence L(Ω0 ) contains
an open set, but this is in contradiction with Lemma 3.1.
Let Ω ∈ S r and S q (Ω) = (skq , ϕkq )q−1
k=0 be a marked symmetric maximal
periodic orbit of rotation number 1/q; then we can define the functional
`S q which is defined as follows: for any continuous function ν : T1 → R
we let
e_definitionEllq
(3.3)
`S q (ν) :=
q−1
X
k=0
ν(skq ) sin ϕkq
DYNAMICAL SPECTRAL RIGIDITY
11
Remark 3.8. These functionals can, of course, be defined for any periodic orbit (rather than only for marked symmetric maximal orbits).
Since we will not use non-symmetric orbits for the proof, we find simpler to use the above definition.
deform
Proposition 3.9. Let (Ωτ )τ ∈U be an isospectral family: then for any
τ ∈ U , q > 1 and S̄ q a maximal marked periodic orbit for Ωτ , we have
`S̄ q (n(τ, ·)) = 0.
Proof. Since the family (Ωτ )τ ∈U is isospectral, by Lemma 3.7 ∆q (τ ) ≡
∆q (0) for any τ ∈ U ; we thus conclude that ∆q is indeed differentiable
and therefore, by Lemma 3.5, for any τ fixed, let ξ¯ realize the maximum,
i.e. ∆q (τ ) = maxξ Lq (τ ; ξ) = Lq (τ, ξ), we have
¯ ∈ D− ∆q (τ ) = {0}.
∂τ Lq (τ ; ξ)
Pq−1
q
k k+1
¯
In particular for S̄ q = (s̄kq , ϕ̄kq )q−1
)
k=0 , since L (τ ; ξ) =
k=0 Lτ (s̄q , s̄q
and observing that
∂τ Lτ (s, s0 ) = ∂τ kγτ (s) − γτ (s0 )k
=
γτ (s) − γτ (s0 )
· [∂τ γτ (s) − ∂τ γτ (s0 )]
0
kγτ (s) − γτ (s )k
we get
¯ = ∂τ
0 = ∂τ L (τ ; ξ)
q
q−1
X
Lτ (s̄kq , s̄k+1
)
q
k=0
q−1
=
"
X
k=0
=
q−1
X
γτ (s̄kq )
kγτ (s̄kq )
#
− γτ (s̄k−1
)
γτ (s̄k+1
) − γτ (s̄kq )
q
q
−
· ∂τ γτ (s̄kq )
k−1
k+1
k
− γτ (s̄q )k kγτ (s̄q ) − γτ (s̄q )k
2 sin ϕ̄kq Nτ (s̄kq ) · ∂τ γτ (s̄kq )
k=0
=2
q−1
X
nτ (s̄kq ) sin ϕ̄kq ,
k=0
which concludes the proof.
Remark 3.10. Let S = {(τ, ∂τ Lq (τ ; ξ))|∂ξ Lq (τ, ξ) = 0}, then S ⊂ T ∗ U
is a Lagrangian submanifold. Let π : T ∗ U → U denote the natural
projection, then Lq (t; ·) is a Morse function (critical points are nondegenerate) if and only if τ is a regular value of the map π|S . The
set U1 of such values is an open set, and by Sard’s theorem, it has
full measure. Furthermore, the set U0 ⊂ U1 of τ such that Lq (τ ; ·)
is an excellent Morse function (Morse function whose critical points
12
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
have pairwise distinguished critical values) is also an open subset of
full measure. For any τ0 ∈ X0 , the critical points of Lq (τ ; ·) depends
smoothly on τ within a sufficiently small neighbourhood of τ0 . Hence
we have
d
∆q (τ ) = `S q (nτ ), τ ∈ U0
dτ
for an arbitrary C 2 deformation γτ (not necessarily isospectral).
Given a symmetric domain Ω ∈ S r and having fixed a choice of
marked maximal symmetric periodic orbits {S q }q≥2 , we will use the
shorthand notation `q = `S q . We now proceed to define two additional
functionals, which we denote with `0 and `1 : `0 (ν) provides the infinitesimal change in the perimeter of Ω due to an orthogonal deformation
generated by ν.
It is not difficult to find an explicit formula for `0 : let Γ(s) be an
arc-length parametrization of ∂Ω, and let δΓ(s) be an infinitesimal
deformation; then we have:
Z l∂Ω
l∂Ω =
kΓ0 (s)kds.
0
Hence, computing the variation and integrating by parts:
Z l∂Ω
Z l∂Ω
0
0
δl∂Ω =
Γ (s) · δΓ (s)ds = −
Γ00 (s) · δΓ(s)
0
0
Z l∂Ω
N (s) · δΓ(s)
=
ds.
ρ(s)
0
e_definitionellp
perimeterIsospectral
We conclude that, if s is the arc-length parametrization of ∂Ω, then
`0 (ν) can be obtained by the formula:
Z l∂Ω
ν(s)
(3.4)
`0 (ν) =
ds
ρ(s)
0
Remark 3.11. Observe that if (Ωτ )τ ∈U is an isospectral family, we
have that, for any Ωτ , the corresponding functional `0 (nτ ) = 0. This
follows by a continuity argument analogous to the one which holds for
periodic orbits and Lemma 3.1.
We now proceed to define `1 (ν) as the evaluation of the perturbation
ν at the marked point s = 0, that is we simply let
`1 (ν) = ν(0).
Observe that if ν preserves our normalization condition, the marked
point is fixed at the origin and, therefore, `1 (ν) = 0.
DYNAMICAL SPECTRAL RIGIDITY
13
Let us now define the space of C r -smooth even functions
r
Csym
= {ν ∈ C r (T1 ) s.t. ν(s) = ν(−s)}.
r
We then define the linearized isospectral operator T : Csym
→ RN :
T ν = (`0 (ν), `1 (ν), · · · , `q (ν), · · · )
In fact, T has range in `∞ , by definition of the functionals `q , since
by [2, Lemma 8], sin ϕkq ≤ C/q for some C uniform in q.
Our Main Theorem then follows from
t_main
Theorem 3.12. If r is sufficiently large and Ω ∈ S r is sufficiently
r
close to a circle, the operator T : Csym
→ `∞ is injective.
Proof of Theorem 2.13. Assume that any element of the family (Ωτ )τ ∈U
is sufficiently close to the circle in the sense of Theorem 3.12. Suppose
by contradiction that for some τ ∈ U , the normal deformation n(τ, ·)
is not zero. Then, since T is injective, we gather that there exists q so
that `q (n(τ, ·)) 6= 0; this contradicts Proposition 3.9.
The rest of this paper is devoted to the proof of Theorem 3.12.
4. A convenient coordinate system
In order to investigate the properties of the functionals `q ’s it is
convenient to introduce a change of variables, which conjugates the
dynamics to a simpler normal form. Such coordinates are of course
closely related to the Lazutkin coordinates (see [12]) and will be obtained via an approximation scheme.
l_Lazutkin
Lemma 4.1. If δ is sufficiently small and r sufficiently large, for any
Ω ∈ Dr δ-close to a circle:
(1) there exist C r−7 coordinates (x̄, ȳ) which conjugate the billiard
map to (x̄, ȳ) 7→ (x̄∗ , ȳ ∗ ), where
x̄∗ = x̄ + ȳ + O(δy 6 )
ȳ ∗ = ȳ + O(δy 6 )
where O(ξ) denotes a function whose sup-norm tends to 0 at
least as fast as ξ.
(2) there exist C r−5 functions α̃(x) and β̃(x) so that kα̃k, kβ̃k =
O(δ) and
x = x̄ + α̃(x)ȳ 2
ϕ = µ(x)ȳ + β̃(x)ȳ 3
14
e_lazutkin
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
where x is the standard Lazutkin parameterization
Z
−1
Z s
−2/3
−2/3
(4.1) x(s) = CΩ
ρ(ξ)
dξ, where CΩ =
ρ(ξ)
dξ
.
0
∂Ω
and
correction-function
odicLazutkinDynamics
(4.2)
µ(x) =
1
,
2CΩ ρ(x)1/3
The proof of Lemma 4.1 will be given in Appendix A.2.
From the first item of Lemma 4.1 we conclude that if ((x̄kq , ȳqk ))k∈{0,··· ,q−1}
is a periodic orbit of rotation number 1/q, we have:
x̄kq = x̄0q + k/q + O(δq −4 )
ȳqk = 1/q + O(δq −5 ).
which in turn imply that there exists C r−5 function α, β and β̃˜ so that
α(x0q + k/q) − α(x0q )
+ O(δq −4 )
2
q
˜
µ(x0q + k/q) β̃(x0q + k/q)
ϕkq =
+
+ O(δq −5 )
q
q3
xkq = x0q + k/q +
which finally yields:
sin ϕkq
µ(x0q + k/q)
β0 (x0q + k/q)
= sin
+
+ O(δq −5 )
q
q3
β(x0q + k/q)
wq
0
= µ(xq + k/q)
1+
+ O(δq −5 )
q
q2
where we defined for q ≥ 2.
wq =
q
π
sin
π
q
Since we will apply the above formulas to marked maximal symmetric
periodic orbits, we can set x0q = 0 and obtain the simplified expressions
(observe that by normalization α(0) = 0);
(4.3a)
(4.3b)
α(k/q)
+ O(δq −4 )
q2
wq
β(k/q)
k
sin ϕq = µ(k/q)
1+
+ O(δq −5 )
q
q2
xkq = k/q +
DYNAMICAL SPECTRAL RIGIDITY
alphabeta
15
Lemma 4.2. If Ω ∈ S r , we have that α(x) is odd, and β(x) is even.
In particular, we have αk = −α−k is purely imaginary, βk = β−k is real:
X
X
α(x) =
2iαk sin 2πkx, β(x) =
2βk cos 2πkx
k≥1
k≥0
where αk and βk are Fourier coefficients of the functions α and β,
respectively.
Proof. The proof is immediate from the form of (4.3).
5. Functionals in Lazutkin coordinates
We now proceed to investigate the form of the functionals `q ’s in the
Lazutkin coordinate obtained in the previous section. Let us introduce
the Lazutkin weighting operator W : C 0 (∂Ω) → C 0 (T1 ) given by:
[Wn](x) = n(s(x))µ(x)
The operator W is injective; let us denote
`˜q (u) = `q (u(x(s))µ(x(s))−1 )
and correspondingly
T̃ (u) = `˜0 (u), · · · , `˜q (u), · · · .
Due to the explicit formulas (4.3), the functionals `˜q ’s turn out to be
simpler to study than `q . First, observe that changing variable in the
integral in the definition (3.4) (using the explicit formula (4.1)), we
obtain
Z 1
`0 (n) =
[Wn](x)dx,
0
hence `˜0 is simply the functional which computes the average with
respect to the Lebesgue measure. `˜1 is the evaluation at the marked
point. In order to obtain a useful expression for `˜q for q ≥ 2, recall the
definition
q−1
X
`q (n) =
n(skq ) sin(ϕkq )
k=0
((skq , ϕkq ))k∈{0,··· ,q−1}
e_tlfuncExpression
where
is a marked maximal symmetric periodic
orbit. Using (4.3) we conclude:
q−1 1X
α(k/q)
β(k/q)
˜
(5.1) `q (u) = wq
u k/q +
1+
+ O(δq −4 )
q k=0
q2
q2
16
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Let us now define the Fourier basis B = (ej )j≥0 of even functions on
the circle in Lazutkin parametrization:
ej = cos 2πjx
j ≥ 0.
Let us introduce the convenient notation
(
1 if q|j
δq|j =
0 otherwise
lem:l(e)-expansion
Lemma 5.1. For any Ω there exists a functional `˜∗ so that for any
q ≥ 2, j ≥ 1:
`˜∗ (ej )
β0
−1
˜
`q (ej )wq =
+ 1 + 2 δq|j +
q2
q
X
1
+
(βsq−j + 2πijαsq−j ) + O(δj 2 q −4 )
q2
s∈Z\{0}
sq6=j
where `˜∗ (ej ) = βj −2πijαj , and αj , βj are Fourier coefficients in Lemma
4.2.
Proof. The lemma follows from a simple computation: by plugging in
u = ej in (5.1) we obtain:
`˜q (ej )wq−1
q−1
1X
α(k/q)
β(k/q)
=
cos 2πj k/q +
1+
+ O(δq −4 )
2
2
q k=0
q
q
q−1 1X
α(k/q)
β(k/q)
=
cos(2πjk/q) − 2πj sin(2πjk/q)
1+
q k=0
q2
q2
+ O(δj 2 q −4 )
= δq|j
"
q−1
1 XX
+ 3
2πij(exp(2πijk/q) − exp(−2πijk/q))αp exp(2πipk/q)+
2q k=0 p∈Z
#
+ (exp(2πijk/q) + exp(−2πijk/q))βp exp(2πipk/q)
+ O(δj 2 q −4 )
1 X
= δq|j + 2
[βsq−j + 2πijαsq−j + βsq+j − 2πijαsq+j ] + O(δj 2 q −4 )
2q s∈Z
DYNAMICAL SPECTRAL RIGIDITY
17
and using the fact that αp = −α−p and βp = β−p
1 X
= δq|j + 2
[βsq−j + 2πijαsq−j ] + O(δj 2 q −4 )
q s∈Z
β0
βj − 2πijαj
= 1 + 2 δq|j +
+
q
q2
1 X
+ 2
(βsq−j + 2πijαsq−j ) + O(δj 2 q −4 ).
q
s∈Z\{0}
sq6=j
which proves our lemma setting `˜∗ (ej ) = βj − 2πijαj .
6. Proof of Theorem 3.12
r
r
First of all let us introduce the projector P∗ : Csym
→ C∗,sym
, where
r
1
is the space of real, even, zero average C -functions of T :
Z
P∗ u = u − u(x)dx.
r
C∗,sym
Let L1∗,sym be the space of L1 , even, zero average, functions of the
circle, i.e.
Z
1
1
1
L∗,sym = u ∈ L (T ) s.t. u(x) = u(−x), u(x)dx = 0
(
)
X
1
1
= u ∈ L (T ) s.t. u(x) =
ûj ej .
j≥1
where ûk denote its Fourier coefficients in the basis B.
We now proceed to define a space of admissible functions: for 3 <
γ < 4, define the subspace
X∗,γ = {u ∈ L1∗,sym s.t. lim j γ |ûj | = 0};
j→∞
equipped with the norm:
kukγ = max j γ |ûj |.
j≥1
The space (X∗,γ , k · kγ ) is a (separable) Banach space.
dγe−1
bγc−1
Remark 6.1. By definition C∗,sym ⊂ X∗,γ ⊂ C∗,sym , whence the functionals
!
∞
∞
X
X
`˜q (n) = `˜q
nj e j =
nj `˜q (ej ).
j=1
j=1
are well-defined on X∗,γ , since the Fourier series converges uniformly.
18
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
We will choose γ > 3. We will always assume r > γ.
∞
Let `∞
s.t. a0 = 0} and let us introduce the
∗ = {b = (ai )i≥0 ∈ `
subspace
γ
h∗,γ = {b = (ai )i≥0 ∈ `∞
∗ s.t. lim j aj = 0}
j→∞
equipped with the norm |b|γ = maxj≥0 j γ |aj |;
l_decomposition
Lemma 6.2. . There exist linearly independent vectors b1 , b∗ ∈ `∞ \
r
h∗,γ so that, for any Ω ∈ S r with r sufficiently large, T̃ : Csym
→ `∞
can be decomposed as follows:
h
i
T̃ = bl `˜0 + b∗ `˜∗ + T̃∗,R P∗
where T̃∗,R : X∗,γ → h∗,γ is an invertible operator.
Proof of Theorem 3.12. By assumptions, the vectors bl , b∗ and T̃∗,R P∗ u
r
are linearly independent for any u ∈ Csym
. Hence, if u ∈ ker T̃ , then
˜
˜
necessarily `0 (u) = `∗ (u) = 0 and T̃∗,R P∗ (u) = 0. Now, by definition, if
r
`˜0 (u) = 0, then u = P∗ u ∈ C∗,sym
. Since T̃∗,R is injective and T̃∗,R u = 0
we thus conclude that u = 0.
Proof of Lemma 6.2. Let us first decompose
T̃ = T̃ ((1 − P∗ ) + P∗ ) = T̃ (1 − P∗ ) + T̃∗ P∗
r
where T̃∗ is the restriction of T̃ on C∗,sym
. Observe that, by definition
(1 − P∗ )u is the constant function equal to `˜0 (u); we can thus set
bl := T̃ (1) = (1, 1, w2 + O(δ), · · · , wq + O(δq −2 ), · · · ) ∈ `∞ .
We thus conclude that T̃ = bl `˜0 + T̃∗ P∗ . Observe moreover that by
Lemma 5.1 we can write T̃∗ = b∗ `˜∗ + T̃∗,R , where
b∗ = (0, 0, w2 /4, · · · , wq /q 2 , · · · )
and recall that
X
X
`˜∗ (u) = `˜∗ (
ûj ej ) =
ûj (βj − 2πjαj )
j≥1
j≥1
`˜∗ (u) seems to define on the projection space X∗,γ .
Clearly, b∗ and bl are linearly independent and do not belong to h∗,γ
since γ ≥ 2. On the other hand, let T̃qj denote the matrix representation of T̃∗,R in the canonical bases.
DYNAMICAL SPECTRAL RIGIDITY
19
For q = 1, we have by definition T̃1j = `˜1 (ej ) = 1; on the other hand,
for q ≥ 2, Lemma 5.1 yields the expression:
#
"
β0
1 X
T̃qj = wq 1 + 2
δq|j + 2
(βsq−j + 2πijαsq−j ) + O(δj 2 q −4 )
q
q
s∈Z\{0}
sq6=j
β0
= wq 1 + 2 [∆qj + Rqj ]
q
where ∆qj = δq|j and Rqj is the remainder. Clearly, it suffices to show
that [∆qj + Rqj ] is invertible provided that δ is sufficiently small. We
will in fact show that
e_invertibility
k∆ + R − Idkγ < 1.
(6.1)
where k · kγ is the operator norm from (X∗,γ , k · kγ ) to (h∗,γ , | · |γ ). The
first step is to establish the following
Lemma 6.3. The operator ∆ : X∗,γ → h∗,γ , identified by the matrix
∆qj is such that k∆ − Idkγ < 2/3; in particular ∆ has bounded inverse.
Proof. By definition the norm k∆ − Idkγ is given6 by:
"
#
X
X
k∆ − Idkγ = sup q γ
(j −γ δq|j − δqj ) = sup q γ
j −γ δq|j − 1
q
= sup q γ
q
q
j>0
X
(sq)−γ − 1 ≤
s>0
X
s>0
since γ > 2.
j>0
s−2 − 1 =
π2
−1
6
In order to conclude the proof of (6.1) it thus suffices to show that
by taking δ sufficiently small, we can make kRqj kγ < 1/3.
First, we check the term O(δj 2 q −4 ); we obtain
X
O(δ) sup
O(j 2−γ q γ−4 ).
q
j>0
We thus need γ > 3 for the sum on j to converge, and γ < 4 for the
sequence to converge to 0 as q → ∞. We conclude that this term can
be made as small as needed by choosing δ sufficiently small.
6
here δqj is the usual Kronecker delta notation.
20
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Next, we deal with the sum: observe that, since α and β are r0 = r−5
smooth
X
X
1
γ −γ 1 γ−2
q j
(β
+
2πijα
)
<
O(δ)q
sq−j
sq−j 2
q |sq − j|r0 j γ−1
s∈Z\{0}
s∈Z\{0}
sq6=j
sq6=j
Where we can choose r0 as large as needed by choosing r large. Let us
now estimate once again the sum over j:


XX
X
X
X
1
1
γ−2


q γ−2
+
= I + II
0 γ−1 = q
r
|sq − j| j
|sq − j|r0 j γ−1
+
+
s j>0
s
0<j<[sq]
j>[sq]
where [sq]+ = max{0, sq}. Then
X 1 X
1
C X 1−γ C
II ≤ q γ−2
=
|s|
<
|sq|γ−1 j>sq |sq − j|r0
q s
q
s
then we write:

I = q γ−2
X

X

s
0<j<sq/2
+
X

sq/2≤j<sq
1
= I0 + I00
|sq − j|r0 j γ−1
Then
0
I ≤q
γ−2
X
X
s>0 0<j<sq/2
I00 ≤ q γ−2
X
0
2r
1
C XX 1
C
<
0
0 1+γ <
r
γ−1
r−γ
r
r−γ
|sq| j
q
s j
q
s>0 j>0
X
s>0 sq/2<j<sq
1
2γ−1
C XX 1
C
<
0
0 γ−1 <
r
γ−1
r
|sq − j| (sq)
q s>0 p>0 p s
q
which allows to conclude provided γ < r0 .
Appendix A. Lazutkin coordinates
s_Lazutkin
A.1. An abstract setting. Let F : S1 × S1 → S1 × S1 be a monotone orientation preserving twist C s diffeomorphism with an involution
given by I : (x, y) 7→ (x, −y), (that is I ◦F −1 = F ◦I). Observe in particular that F fixes the circle y = 0, that is F (x, 0) = (x, 0). Assume
moreover that the lift of F to R2 satisfies the condition F (x, y + 2π) =
(x + 2π, y + 2π).
Then let us introduce a convenient definition: for N ≥ 0 consider a
solution L : C s (S1 × S1 , S1 ) of the following equation:
e_lazutkinFunction
(A.1)
L ◦ F − L = L − L ◦ F −1 + O(y N ).
nFunctionSymmetrized
e_LazutkinNormalForm
DYNAMICAL SPECTRAL RIGIDITY
21
Using the properties of I we can rewrite (A.1) as
L ◦ F + L ◦ I ◦ F ◦ I − L = O(ϕN ).
Observe now that if L solves the above equation, then L ◦ I is also a
solution. By linearity of the equation, we further conclude that L can
always assumed to be invariant under I, that is, an even function of y.
Given a function L, we can define its even and odd part (with respect
to the second coordinate) as:
1
1
[L]even = [L + L ◦ I]
[L]odd = [L − L ◦ I]
2
2
Such function will, thus, satisfy the following symmetrized coboundary
equation:
(A.2)
[L ◦ F ]even − L = O(y 2K )
where we set N = 2K since the left hand side is an even function of y.
An even function L ∈ C s (S1 × S1 , S1 ) is called a Lazutkin function
of order 2K if
• for any fixed y ∈ S1 the map L(·, y) : S1 → S1 is an orientationpreserving diffeomorphism.
• L satisfies (A.2).
Let us now define YL (x, y) as the following real function, odd in y:
YL = [L ◦ F ]odd
Observe that since F is a twist map, YL (x, ·) is strictly increasing
for any fixed x. Moreover, by our assumptions on F we gather that
YL (x, 2π) = YL (x, 0) + 2π. We conclude that, for any x ∈ S1 , the function YL (x, ·) on a diffeomorphism yL (x, ·) : S1 → S1 In other words,
setting xL (x, y) = L(x, y) we have that (x, y) 7→ (xL , yL ) is a nondegenerate change of coordinates. It is simple to see that in these
coordinates the map F reads
(
2K
xL 7→ xL + yL + r0 (xL , yL )yL
(A.3)
FL :
2K
yL 7→ yL + r1 (xL , yL )yL
where r0 , r1 ∈ C s (S1 × S1 ) and, moreover:
2K
r0 (xL , yL )yL
= [L ◦ F (x, y)]even − L(x, y)
r1 (xL , yL ) = 2r0 (xL , yL ) + O(yL )
and (x, y) are seen as function of (xL , yL ). Moreover, since yL is an
odd function of y, we gather that the involution has the same form
I : (xL , yL ) 7→ (xL , −yL ) in such coordinates. In particular, r0 is an
even function of y. We call (xL , yL ) Lazutkin variables of order 2K.
22
r_canonicalForm
e_canonicalTwist
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
Remark A.1. Observe that since F fixes the circle y = 0, L(x, y) = x
is a Lazutkin function7 of order 2, provided F ∈ C 3 . In particular,
replacing y with [F (x, y) − x]odd = ∂y x∗ (x, 0)y + O(y 3 ) we can always
assume that F is given the canonical form
(
x 7→ x + y + r0 (x, y)y 2
(A.4)
F :
y 7→ y + r1 (x, y)y 2 .
The following lemma constitutes the main result of this section. It
states that provided that r0 and r1 are sufficiently small and sufficiently
smooth, then we can always find Lazutkin coordinates of higher order.
l_lazutkinFunctions
Lemma A.2. Assume that for some K > 0 and s ≥ 3, there exist C s
functions r0 , r1 so that:
(
x 7→ x + y + r0 (x, y)y 2K
F :
y 7→ y + r1 (x, y)y 2K
then if kr0 kC 0 is sufficiently small8, there exists a C s change of coordinates (x, y) 7→ (x̄, ȳ) and C s−2 functions r̄0 , r̄1 so that
(
x̄ 7→ x̄ + ȳ + r̄0 (x̄, ȳ)ȳ 2K+2
F̄ :
ȳ 7→ ȳ + r̄1 (x̄, ȳ)ȳ 2K+2 .
Moreover for i ∈ {0, 1}, we have kr̄i kC s−2 ≤ C(kr0 kC s + kr1 kC s ).
Proof. The key to the proof is to find suitable Lazutkin functions; to
simplify the exposition it is convenient to treat separately the case
K = 1 and K > 1.
Let us first consider the case K = 1 (which corresponds to (A.4));
then we look for a Lazutkin function L of order 4. We will make the
ansatz that L(x, y) = l(x), where l solves the differential equation
e_ansatz1
(A.5)
2l0 (x)r0 (x, 0) + l00 (x) = 0
with boundary conditions l(0) = 0 and l(2π) = 2π. Let us prove that
L is a Lazutkin function: first of all, ODE considerations imply that
l0 > 0, which, together with the boundary conditions, implies that
x 7→ L(x, y) is a circle diffeomorphism for any y. We thus need to
show that L satisfies (A.2). First of all observe that we can write
r0 (x, y) = r0 (x, 0) + r̃0 (x, y)y 2
7 As a matter of fact,
any diffeomorphism independent of ϕ would give a Lazutkin
function of order 2.
8 This assumption is actually not needed if K = 1
DYNAMICAL SPECTRAL RIGIDITY
23
where r̃0 is a C s−2 function. For simplicity, let F (x, y) = (x∗ , y ∗ ); then
we can write
l00 (x) 2
l(x∗ ) = l(x) + l0 (x)[y + r0 (x, 0)y 2 ] +
y + l̃(x, y)y 3
2
l00 (x)
0
0
= l(x) + l (x)y + l (x)r0 (x, 0) +
y 2 + l̃(x, y)y 3
2
and by (A.5) we conclude
= l(x) + l0 (x)y + l̃(x, y)y 3
where l̃ is C s−2 . Hence (A.2) is satisfied with a C s−2 remainder that
we denote with r̄0 . The estimates on the norms are immediate. The
case K > 1 is similar, but simpler: this time the ansatz is L(x, y) =
x + l(x)y 2(K−1) , where we assume that
e_anzatzK
(A.6)
l00 (x) = −2r0 (x, 0)
with boundary conditions l(0) = l(2π) = 0. Then, if kr0 kC 0 is sufficiently small ∂x L(x, y) = 1 + l0 (x)y 2 > 0, thus L(·, y) is a circle
diffeomorphism. Moreover, again we can write:
L(x∗ , y ∗ ) = x + y + r0 (x, 0)y 2K + l(x)y 2(K−1)
l00 (x) 2K
y + l̃(x, y)y 2K+1
2
= x + l(x)y 2(K−1) + y
l00 (x)
0
2K−1
+ l (x)y
+ r0 (x, 0) +
y 2K + l̃(x, y)y 2K+1
2
+ l0 (x)y 2K−1 +
and again using our ansatz
= L(x, y) + y + l0 (x)y 2K−1 + l̃(x, y)y 2K+1 .
We thus conclude, by taking the even part, that L(x, y) is a Lazutkin
function of order 2K + 2 with C s−2 remainder which we call r̄0 (x, y).
Once again, the estimates on the norms are immediate.
s_lazutkin
A.2. Billiards: the proof of Lemma 4.1. We now proceed to apply
the result of the previous section to the case of billiard dynamics
Let us fix a domain Ω and denote with (s+ (s, ϕ), ϕ+ (s, ϕ)) (resp.
−
(s (s, ϕ), ϕ− (s, ϕ))) the coordinates of the image (resp. preimage) of
(s, ϕ) under the billiard map f = fΩ , where recall that s is the arclength parameter on ∂Ω and ϕ ∈ [0, π] is the angle that the outgoing
trajectory forms with the positively oriented tangent at s. It follows
24
JACOPO DE SIMOI, VADIM KALOSHIN, AND QIAOLING WEI
from the definition that the map I(s, ϕ) 7→ (s, π − ϕ) is an involution
for the billiard dynamics, that is I ◦ f −1 = f ◦ I, or more explicitly
(s− (s, ϕ), π − ϕ− (s, ϕ)) = s+ (s, π − ϕ), ϕ+ (s, π − ϕ).
For convenience, we will consider the angle ϕ to be defined on S1 , rather
than on [0, π], via the identification ϕ ∼ ϕ + π. Then, by the properties
of the involution we have:
s− (s, ϕ) = s+ (s, π − ϕ) = s+ (s, −ϕ)
ϕ− (s, ϕ) = π − ϕ+ (s, π − ϕ) = −ϕ+ (s, −ϕ)
For symmetry of notation, let us now denote s∗ = s+ and ϕ∗ = ϕ+ . It
follows from simple geometrical observations that s∗ and ϕ∗ are both
smooth at ϕ = 0.
As described in Remark A.1, we can put the billiard map in the
canonical form (A.4) by choosing x = s, which gives coordinate reads:
y(s, ϕ) = 2ρ(s)ϕ + O(ϕ2 )
r_circle
e_lazutkinApp
Remark A.3. Notice that if ∂Ω is a circle, the ρ(s) is constant equal
to the radius of the circle and the billiard map reduces to:
(
x 7→ x + y
(A.7)
f:
y 7→ y
By following the procedure described in the proof of Lemma A.2, we
can obtain Lazutkin coordinates of order 4 by means of the change of
variables
(A.8)
Z
−1
Z x
−2/3
−2/3
x̄(x) = l(x) = CΩ
ρ(ξ)
dξ, where CΩ =
ρ(ξ)
dξ
.
0
∗
and expressing x = x +
∂Ω
P
k bk y
k
, we obtain the explicit formula
l000 3 3
0
0
00
ȳ = l b1 y + l b3 + l b1 b2 +
b y =
6 1
2ρ̇2 − 3ρρ̈ 3
−2/3
= CΩ ρ
y+
y
108ρ2
2ρ̇2 − 3ρρ̈ 3
1/3
= 2CΩ ρ
ϕ+
ϕ
27
Notice that, while the x̄ coordinate is the same as the one used by
Lazutkin (see [12]), the ȳ coordinate is not (they differ at order ϕ3 ):
in fact in these coordinates the dynamics is conjugated to (A.3) with
K = 2 with remainder term kr0 kkC ≤ Ckρ̇k3C 2+k . Lemma 4.1 then
DYNAMICAL SPECTRAL RIGIDITY
25
follows by applying Lemma A.2 to find Lazutkin coordinates of order
6.
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Jacopo De Simoi, Department of Mathematics, University of Toronto,
40 St George St. Toronto, ON, Canada M5S 2E4
E-mail address: [email protected]
URL: http://www.math.utoronto.ca/jacopods
Vadim Kaloshin, Department of Mathematics, University of Maryland, College Park, 20742 College Park, MD, USA.
E-mail address: [email protected]
URL: http://www.math.umd.edu/~vkaloshi/
Qiaoling Wei, University of Maryland, College Park, 20742 College Park, MD, USA.
E-mail address: [email protected]