We propose a simple approach to a problem introduced by Galatolo and
Pollicott, consisting in perturbing a dynamical system in order for its
absolutely continuous invariant measure to change in a prescribed way. Instead
of using transfer operators, we observe that restricting to an infinitesimal
conjugacy already yields a solution. This allows us to work in any dimension
and dispense from any dynamical hypothesis. In particular, we don't need to
assume hyperbolicity to obtain a solution, although expansion moreover ensures
the existence of an infinite-dimensional space of solutions.Comment: v2: the approach has been further simplified, only basic differential
  calculus is in fact needed instead of basic PD

Kloeckner, Benoît

English

arXiv

v2: the approach has been further simplified, only basic differential calculus is in fact needed  instead of basic PDE.We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yields a solution. This allows us to work in any dimension and dispense from any dynamical hypothesis. In particular, we don’t need to assume hyperbolicity to obtain a solution, although expansion moreover ensures the existence of an infinite-dimensional space of solutions

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The linear request problem

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The linear request problem
Benoît Kloeckner
To cite this version:
Benoît Kloeckner. The linear request problem. 2017. ￿hal-01327440v2￿
The linear request problem
Benoit R. Kloeckner∗†
†
Université Paris-Est, Laboratoire d’Analyse et de Matématiques Appliquées (UMR 8050),
UPEM, UPEC, CNRS, F-94010, Créteil, France
June 23, 2017
We propose a simple approach to a problem introduced by Galatolo and
Pollicott, consisting in perturbing a dynamical system in order for its abso-
lutely continuous invariant measure to change in a prescribed way. Instead
of using transfer operators, we observe that restricting to an infinitesimal
conjugacy already yields a solution. This allows us to work in any dimension
and dispense from any dynamical hypothesis. In particular, we don’t need
to assume hyperbolicity to obtain a solution, although expansion moreover
ensures the existence of an infinite-dimensional space of solutions.
Let M be a compact n-manifold and T : M → M be a smooth map, seen as a discrete-
time dynamical system. Assume that there is a positive smooth invariant measure ω,
also seen as a nowhere vanishing n-form.
The linear response theory(see e.g. [Rue09, BS12]) is typically concerned with the
following question: if (Tt)t∈(−ε,ε) is a family of maps with T0 = T , differentiable at t = 0,
such that each Tt preserves a smooth measure ωt, can we differentiate ωt with respect
to t and express
.
ω0 :=
d
dt
∣
∣
t=0
ωt in term of
.
T0 :=
d
dt
∣
∣
t=0
Tt?
A recent article by Galatolo and Pollicott [GP16] proposes to study the opposite
direction; they roughly ask two questions (and provide some answers for expanding
maps of the circle), the second of which we voluntarily keep imprecise:
Question 1. Given a smooth function ρ : M → R of vanishing integral,1 can we find a
perturbation (Tt)t∈(−ε,ε) of T , differentiable at t = 0 and preserving a family of smooth
measure (ωt)t∈(−ε,ε) such that
.
ω0 = ρω? Can we then express the possible values of
.
T0
in term of ρ?
∗Supported by the Agence Nationale de la Recherche, grant ANR-11-JS01-0011.
1The condition
∫
M
ρω = 0 is needed whenever invariant measures are normalized to have fixed total
mass.
1
Question 2. When the previous question has a positive answer, can we find an “optimal”
value of
.
T0?
The goal of this note is to observe that Question 1 has a positive answer in a very
general setting, without dynamical hypothesis and in every dimension. This follows
from well-known facts in differential geometry, and the only faint bit of novelty is in the
observation that one can restrict to conjugate deformations, of the form Tt = ϕ
t◦T ◦ϕ−t
where (ϕt)t∈(−ε,ε) is the flow of a vector field.
Let us first define properly what it means for a family (Tt)t∈(−ε,ε) to be differentiable
at t = 0. Galatolo and Pollicott work on the circle and implicitly use its parallelism (all
tangent spaces can be identified), which is not possible on a general manifold. Pointwise,
we want to ask that for each x ∈ M , the curve (Tt(x))t∈(−ε,ε) is differentiable at t = 0,
and we want the derivative of this curve to depend smoothly on x. We want to stress
that
.
T0(x) :=
d
dt
∣
∣
t=0
Tt(x) is an element of TT (x)M , not of TxM , and
.
T0 is thus not a
vector field. We will thus consider the set ΓT (M) of smooth maps Z : M → TM such
that for all x ∈ M , Zx ∈ TT (x)M , and say that a family (Tt)t∈(−ε,ε) of smooth maps
M → M is differentiable at t = 0 if
.
T0(x) is defined for all x and
.
T0 ∈ ΓT (M).
For example, if X is a smooth vector field on M , we can consider its flow (ϕt)t∈R
and the family Tt := ϕ
t ◦ T ◦ ϕ−t. Then a direct computation shows that (Tt)t∈R is
differentiable at 0:
ϕt ◦ T ◦ ϕ−t(x) = ϕt ◦ T
(
x− tXx + o(t)
)
= ϕt
(
T (x)− tDxT (Xx) + o(t)
)
= T (x)− tDxT (Xx) + tXT (x) + o(t)
so that
.
T0(x) = −DxT (Xx) +XT (x), which we also write
.
T0 = −DT (X) +XT . This is
naturally an element of ΓT (M), as it should.
Observe that since T preserves ω, Tt then preserves ωt := ϕ
t
∗
ω. But
d
dt
∣
∣
∣
∣
t=0
ϕt
∗
ω = −LXω
where L denotes the Lie derivative. To answer Question 1 by restricting to such defor-
mations by conjugacy, we thus only have to check that for all ρ such
∫
M
ρω = 0, there
is a vector field X such that LXω = −ρω. This is well-known, see e.g. the proof of
Moser’s theorem in [KH95] (Theorem 5.1.27 page 195), but let us recall the argument
for the sake of completeness.
Since ω is a n-form, the Cartan formula reads LXω = d(iXω) where iXω is the
(n − 1)-form ω(X ; ·; · · · ; ·) obtained by contraction with X. Since M is compact, its
top-dimensional cohomology is 1-dimensional, generated by the class of ω (or any volume
form), and every n-form of vanishing total integral is exact. This means that there exist
a (n− 1) form θ such that dθ = −ρω. Now, ω being a volume form it is non-degenerate,
which means that any (n−1)-form can be obtained by contracting ω with a well-chosen
2
vector field; in particular, there must exist a vector field X such that iXω = θ. Putting
all this together, we get the desired conclusion:
.
ωt = −LXω = − d(iXω) = −dθ = ρω.
We conclude:
Theorem 3. Let T : M → M be a smooth map acting on a compact smooth Riemannian
manifold, preserving a smooth volume form ω. Let ρ : M → R be a smooth function
such that
∫
M
ρω = 0.
There exist a deformation (Tt)(−ε,ε) of T that is differentiable at t = 0 and preserving
smooth volume forms ωt, such that
.
ω0 = ρω. Moreover one can ask all Tt to be smoothly
conjugate to T .
Now, observe that in the above θ is not unique: one can add to it any closed (n− 1)
form, and will obtain a different suitable X. If n ≥ 2, the space of closed (n− 1) form
is infinite-dimensional (it contains all exact forms dα where α is a (n− 2)-form); but if
M = S1, there are no (n− 2)-forms and the space of closed (n− 1)-form is the space of
functions with vanishing derivative, i.e. of constant functions.
This lack of uniqueness points, in dimension at least 2, to the existence of many
solutions to Question 1 even when restricting to conjugate deformations; however, since
we consider perturbations only at their first order, we have to determine whether the
space of possible
.
T0 is indeed as large as the set of vector fields X generating the
conjugacy.
We thus consider smooth vector fields X and Y on M , such that LXω = LY ω = −ρω
and −DT (X) + XT = −DT (Y ) + YT . Does this imply that X = Y ? The answer
strongly depends on T . The first equation means that X − Y preserves ω, while the
second equation can be rewritten as
XT − YT = DT (X − Y ),
i.e. the vector field X − Y is T -invariant. The difference between the spaces of possible
values for X and for
.
T0 above is therefore determined by the space of T -invariant vector
fields preserving ω. There is no general rule: there could be no non-zero such vector
fields (e.g. if T is expanding) or there could be plenty (e.g. if T is the identity, which
of course is a very degenerate case: we get
.
T0 = 0). Depending on this, there could be
an infinite-dimensional space of possible values for
.
T0 in Theorem 3, or a unique one, or
anything in between.
We conclude with a few remarks.
Remark 4. Sometimes one consider perturbations of the form
.
T0(x) = XT (x) for some
vector field X. This is equivalent to considering
.
T0 ∈ ΓT (M) if T is invertible, but is
less general otherwise as the images of any x, y ∈ M such that T (x) = T (y) would be
asked to be perturbed identically.
Remark 5. Using Moser’s theorem [Mos65] instead of its first-order version, one sees
that for all volume form ω′, there is a conjugate of T that preserves ω′.
3
Remark 6. It could be asked what happens in lower regularity, e.g. Ck. We do not
enter into these details, since the strategy would be the very same, only keeping track
of the available regularity for the various objects ω, θ, X, etc.
Remark 7. One can wonder whether the observation here can shed some light on
Question 2. If one measures optimality by fixing a Riemannian metric on M and trying
to minimize ‖
.
T0‖L2, as Galatolo an Pollicott do in [GP16], then in general the optimal
first-order perturbation is not of the form obtained here by conjugacy. This is easily
seen in the case of the doubling map on the circle.
One can also restrict to conjugate perturbations, and try to minimize ‖X‖L2 where
X is the vector field generating the conjugating diffeomorphisms. Then the problem
directly relates to solving the modified Poisson equation
∇ · (η∇u) = −ρ
where η, ρ are the densities of the invariant measure and of its perturbation with respect
to the Riemannian volume and u is the unknown (then ∇u is the optimal vector field
X). This is an elliptic PDE, whose theory is well established and that admits smooth
solutions. The first version of this note used this to obtain Theorem 3, but not insisting
on finding a gradient vector field can be done even more easily, as above. I wish to thank
Anatole Katok for interesting criticism on the first version of this note, pointing to this
simplification.
References
[BS12] Viviane Baladi and Daniel Smania. Linear response for smooth deformations of
generic nonuniformly hyperbolic unimodal maps. Ann. Sci. Éc. Norm. Supér.
(4), 45(6):861–926 (2013), 2012.
[GP16] Stefano Galatolo and Mark Pollicott. Controlling the statistical properties of
expanding maps. arXiv:1603.07762, 2016.
[KH95] Anatole Katok and Boris Hasselblatt. Introduction to the modern theory of
dynamical systems, volume 54 of Encyclopedia of Mathematics and its Applica-
tions. Cambridge University Press, Cambridge, 1995. With a supplementary
chapter by Katok and Leonardo Mendoza.
[Mos65] Jürgen Moser. On the volume elements on a manifold. Trans. Amer. Math.
Soc., 120:286–294, 1965.
[Rue09] David Ruelle. A review of linear response theory for general differentiable
dynamical systems. Nonlinearity, 22(4):855–870, 2009.
4


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