Posted by: matheuscmss | April 4, 2012

Hakan Eliasson’s talk on a problem of Michel Herman

Last March 30, Hakan Eliasson gave a Colloquium Talk (on joint work with Bassam Fayad and Raphael Krikorian) at UMD opening the activities of the Maryland-Penn State Dynamics Meeting (celebrating the 70th birthday of J. Yorke).

The talk concerned a problem in KAM theory posed by Michel Herman at ICM’98 (Berlin). The general plan for the talk was:

  • 1) Statement of M. Herman’s problem
  • 2) Birkhoff normal forms
  • 3) Generalization of a theorem of H. Russmann
  • 4) KAM “counter-term” theorem
  • 5) Back to M. Herman’s problem
  • 6) A {C^{\infty}} example

Below the fold the reader will find my notes for this (excellent) talk by H. Eliasson. Of course, this goes without saying that any mistakes/errors are entirely my fault and responsibility.

1. A problem by M. Herman

Consider a real-analytic Hamiltonian {H(\phi,r)=\langle\omega_0,r\rangle+O^2(r)(\phi)} (where {O^2(r)(\phi)} is real-analytic of {\phi} and {r} whose first derivative at {r=0} vanishes) obtained by perturbation of (the completely integrable Hamiltonian) {H_0(\phi,r)=\langle\omega_0,r\rangle}. Here, {\phi\in\mathbb{T}^d} are “angle-variables”, {r\in\mathbb{R}^d} are “action-variables”, {r\sim 0}, and the frequency {\omega_0} is assumed to be Diophantine, i.e.,

\displaystyle \omega_0\in DC(\kappa_0,\tau_0):=\{\omega\in\mathbb{R}^d: |\langle k,\omega\rangle|\geq \kappa_0(1/|k|)^{\tau_0} \, \forall \, k\in\mathbb{Z}^d-\{0\}\}

for some {\kappa_0>0}, {\tau_0>d}. For sake of simplicity, we will denote this class of Hamiltonian’s {H} by {\mathcal{C}(\omega_0)}.

The Hamiltonian system associated to {H} is the dynamical system (flow) defined by:

\displaystyle \left\{\begin{array}{l}\stackrel{.}{\phi}=\frac{\partial H}{\partial r} = \omega_0+O(r)(\phi) \\ \stackrel{.}{r} = -\frac{\partial H}{\partial \phi} = O^2(r)(\phi)\end{array}\right.

For {r=0}, we see that {\mathbb{T}^d\times\{r=0\}} is a real-analytic flow-invariant Lagrangian torus with flow {\phi\mapsto\phi+t\omega_0}. This is a prototypical example of a KAM torus (KAM standing for A. Kolmogorov, V. Arnold and J. Moser): in general, a KAM torus is, by definition, a real-analytic flow-invariant Lagrangian torus with flow conjugated (real-analytically) to {\phi\mapsto \phi+t\omega_0}.

Concerning the existence of these objects (KAM torii), we have the classical and celebrated KAM theorem:

Theorem (KAM’60, Pöschel’82). Let {H\in\mathcal{C}(\omega_0)}. If Kolmogorov’s non-degeneracy condition

\displaystyle \det\int_{\mathbb{T}^d}\partial^2_r H(\phi,r=0)d\phi\neq 0

holds, then the set of KAM torii has positive measure and it has density {1} at {\mathbb{T}^d\times\{0\}}

The proof of this result is based on hard (“Nash-Moser type”) implicit function theorems derived from some versions of } (since this is a beautiful chapter in the history of Dynamical Systems, it is likely that we will come back to this topic in this blog in the future).

In his talk at ICM’98 (Berlin), Michel Herman posed the following question (see the notes for his talk):

Is the KAM theorem above true for all {H\in\mathcal{C}(\omega_0)} (i.e., can we remove Kolmogorov’s non-degeneracy condition in the previous theorem)?

The main goal of H. Eliasson’s talk is to give a partial answer to this fundamental question.

2. Birkhoff normal forms

Let {H\in\mathcal{C}(\omega_0)}, say {H(\phi,r)=\langle\omega_0,r\rangle+O^2(r)(\phi)}. The next statement (on the so-called Birkhoff normal forms) says that we can, at least formally, perform an exact (symplectic) change of variables to improve the perturbation term {O^2(r)(\phi)} by getting rid of the dependence on {\phi}:

Proposition 1 Let {H\in\mathcal{C}(\omega_0)}. Then, there are

  • {Z(\phi,r)=(\phi+O(r), r+O^2(r))} analytic on {\phi\in\mathbb{T}^d}, formal in {r}, and symplectic (with respect to {dr\wedge d\phi});
  • {N(r)-\langle\omega_0,r\rangle=O^2(r)}.

such that {H\circ Z(\phi,r)=N(r)}. Moreover,

  • {N=N_H} is unique and it is called the Birkhoff normal form (BNF) of {H}
  • {Z=Z_H} is unique modulo “normalization” and it is called Birkhoff normalization of {H}.

By the seminal works of H. Poincaré and C. Siegel, It is known that {Z_H} is “typically” divergent. For instance, H. Poincaré showed that {Z_H} diverges for topologically typical (generic) {H} and C. Siegel showed that the same is true in a more “metric” (measure-theoretic) sense (in terms of the coefficients of the Taylor expansion of the real-analytic function {H}).

On the other hand, very little is known about {N_H}. For instance, the following questions are open:

  • Can {N_H} be divergent?
  • Can {N_H} be convergent when {Z_H} is divergent?

Among the few facts known about {N_H}, one has the following important result:

Theorem (R. Perez-Marco’03). If {N_H} is divergent for some {H\in\mathcal{C}(\omega_0)}, then {N_H} is divergent for “typical” {H\in\mathcal{C}(\omega_0)}.

In a similar vein, one can consider the set

\displaystyle B_{\omega_0}:=\{N_H: H\in\mathcal{C}(\omega_0)\}\subset\mathbb{R}[[r]]

and one can ask whether this set is “large” or “small”. Recently, A. Bounemoura proved that:

Theorem (A. Bounemoura’11). There exists a prevalent set {\mathcal{G}\subset\mathbb{R}[[r]]} such that if {N_H\in\mathcal{G}}, then {\mathbb{T}^d\times\{0\}} is super-exponentially stable in the sense that the Hamiltonian flow associated to {H} takes a time

\displaystyle t_\varepsilon>e^{e^{(1/\varepsilon)^a}}

to move points in the {\varepsilon}-neighborhood of {\mathbb{T}^d\times\{0\}} outside the {2\varepsilon}-neighborhood of it. Here, {a=a(d)>0} is a constant.

3. Generalization of a theorem of H. Russmann

Still around the problem of convergence or divergence of BNF and Birkhoff normalizations, we have the following classical result of H. Russmann:

Theorem (H. Russmann’67). If {N_H(r)=n_H(\langle\omega_0,r\rangle)} (e.g., {N_H(r)=\langle\omega_0,r\rangle}), then {Z_H} and {N_H} are convergent.

Motivated by this result, one introduces the following notion:

Definition 2 We say that {N_H(r)} is {j}-degenerate (for {0\leq j\leq d-1}) if and only if there are {\omega_0,\dots,\omega_{d-j-1}} linearly independent frequency vectors such that

\displaystyle N_H(r)=n_H(\underbrace{\langle\omega_0,r\rangle, \langle\omega_1,r\rangle, \dots,\langle\omega_{d-j-1},r\rangle}_{d-j})

Remark 1 The assumption of H. Russmann’s theorem correspond to {j=d-1} (i.e., {(d-1)}-degenerate BNF), while it is not hard to check that Kolmogorov’s non-degeneracy condition in KAM theorem implies {j=0} (i.e., {0}-degenerate BNF).

In view of this remark, the statement of the following theorem (generalizing Russmann’s one) becomes “natural” (but, of course, very far from obvious):

Theorem 3 (H. Eliasson, B. Fayad, R. Krikorian’12) If {N_H} is {j}-degenerate, then there exists a real-analytic submanifold of dimension {2d-j} close to {\mathbb{T}^d\times\{0\}} foliated by KAM torii with frequency vector {\omega_0}.

In order to give a sketch of the proof of this theorem, we will need a KAM counter-term theorem.

4. KAM “counter-term” theorem

In this section, we consider frequency vectors {\omega} such that:

  • {\omega\in B_1(\omega_0)\subset\mathbb{R}^d} (here {B_1(x)} is the ball of radius {1} around {x})
  • {\omega\in DC(\kappa,\tau_0)} for some {0<\kappa\leq\kappa_0} (recall that {\omega_0\in DC(\kappa_0,\tau_0)\subset DC(\kappa,\tau_0)} when {\kappa\leq \kappa_0}).

Theorem 4 (H. Eliasson, B. Fayad, R. Krikorian’12) There exists an exact symplectic

\displaystyle Z_{c,\omega}(\phi,r)=(\phi+O^2(r)(c), r+O^2(r)(c)

real-analytic in {\phi\in\mathbb{T}^d} for {r,c} close to {0\in\mathbb{R}^d}, and

\displaystyle \Lambda=\Lambda(c,\omega)=\omega-\omega_0+O(c)

a real-analytic function in {c} and a {C^{\infty}}-function in {\omega\in B_1(\omega_0)} such that

\displaystyle (H+\langle\Lambda,r\rangle)\circ Z_{c,\omega}(\phi,r) = \textrm{constant}+\langle\omega, r-c\rangle + O^2(r-c)+g

where {g} is {C^{\infty}}-flat (i.e., {g} and all its derivatives vanish) at {\omega\in DC(\kappa,\tau_0)}.

In other words, this result says that after “adjusting” the Hamiltonian {H} with the “counter-term” {\langle\Lambda,r\rangle}, we can perform an exact symplectic change of variables to get a nice “normal form” at sufficiently Diophantine frequencies {\omega\in DC(\kappa,\tau_0)}.

Here, the “range” of the change of variables {|r|,|c|\leq \delta(\kappa)} depends on {\kappa\leq\kappa_0} (as well as the big {O} estimates).

In any event, the proof of this result goes through “classical KAM arguments” (in the words of H. Eliasson), and, as we’re going to see now, it is a crucial ingredient in the proof of Theorem 3.

4.1. Proof of Theorem 3 assuming Theorem 4

Starting from the statement of Theorem 4, by an application of the implicit function theorem, it is possible to construct from the equation

\displaystyle \Lambda(c,\omega)=\omega-\omega_0+O(c)=0

a function

\displaystyle \omega=\Omega(c)=\omega_0+O(c)

with

\displaystyle H\circ Z_{c,\Omega(c)}(\phi, r) = \langle\Omega(c), r-c\rangle + O^2(r-c)+g

Furthermore, if {\omega=\Omega(c)\in DC(\kappa, \tau_0)} (i.e., {\Omega} “passes” by Diophantine frequencies), then the {C^{\infty}}jet of the term {g} above vanishes (i.e., {g} is {C^{\infty}}-flat at {\omega\in DC(\kappa,\tau_0)}) and this allows us to deduce the presence of KAM torii (by setting {r=c}).

So, this leads us to study {\Omega(c)}. We know that

  • (a) the {C^{\infty}}jet {\widehat{\Omega}(c):=\mathcal{J}^{\infty}_{c=0}\Omega(c)-\omega_0} of {\Omega(c)-\omega_0} is {\nabla N_H(c)} (by uniqueness of BNF);
  • (b) {\Lambda(c,\Omega(c))=0}.

Now, when the {N_H} is {j}-degenerate, we can find {j} linearly independent unitary vectors {v_1,\dots, v_{d-j}\in S^{d-1}} such that {\partial^k_{v_{i_1},\dots,v_{i_k}} N_H(c)|_{c=0}=0} for all {k\geq 1}, {i_1,\dots, i_k\in \{1,\dots, d-j\}}.

In particular, by item (a) above, we get that

\displaystyle \partial^k_{v_{i_1},\dots,v_{i_k}}\widehat{\Omega}(c)|_{c=0}=0

for all {k\geq 0}.

Therefore, by item (b), we conclude that, for all {k\geq 0},

\displaystyle 0=\partial^k_{v_{i_1},\dots,v_{i_k}}\Lambda(c,\Omega(c))|_{c=0}

\displaystyle = \partial^k_{v_{i_1},\dots,v_{i_k}}\Lambda(c, \omega_0)|_{c=0} + \mathcal{F}(\partial^m_{v_{i_1},\dots,v_{i_m}}\widehat{\Omega}(c)|_{c=0}: 1\leq m\leq k)

where {\mathcal{F}} is a function of the partial derivatives of {\Lambda} and {\partial^m_{v_{i_1},\dots,v_{i_m}}\widehat{\Omega}(c)|_{c=0}} determined by Leibnitz formula.

In our situation, we know (see above) that {\partial^m_{v_{i_1},\dots,v_{i_m}}\widehat{\Omega}(c)|_{c=0}=0} for all {m\geq 1}, so that it is not hard to deduce that {\mathcal{F}(\partial^m_{v_{i_1},\dots,v_{i_m}}\widehat{\Omega}(c)|_{c=0}: 1\leq m\leq k)=0}, and, a fortiori,

\displaystyle 0 = \partial^k_{v_{i_1},\dots,v_{i_k}}\Lambda(c, \omega_0)|_{c=0}

for all {k\geq 0}.

However, since the function {c\mapsto \Lambda(c,\omega_0)} is real-analytic (see the statement of Theorem 4), we conclude that

\displaystyle \Lambda(\sum\limits_{i=1}^{d-j} t_i v_i,\omega_0)=0

for all {t_1,\dots, t_{d-j}\in\mathbb{R}} close to {0\in\mathbb{R}}, that is,

\displaystyle \Omega(\sum\limits_{i=1}^{d-j} t_i v_i)=\omega_0\in DC(\kappa_0,\tau_0)

for all {t_1,\dots, t_{d-j}\in\mathbb{R}} close to {0\in\mathbb{R}}, and, as we mentioned above, this gives a real-analytic submanifold of dimension {d+(d-j)-2d-j} close to {\mathbb{T}^d\times\{0\}} and foliated by KAM torii with frequency vector {\omega_0(=\Omega(\sum\limits_{i=1}^{d-j} t_i v_i))}. Thus, this completes the sketch of proof of Theorem 3.

5. Back to M. Herman’s problem

Actually, in the case {N_H} is non-degenerate, one can use the analyticity (and non-degeneration) properties of {\Omega(c)} to show that, since {\Omega(c)} “starts” at a Diophantine frequency (in the sense that {\omega_0=\Omega(0)}), it must passes through Diophatine frequencies.

In fact, this kind of idea is very well known in the literature (see, e.g., the section “Applications to metric theory of Diophantine approximation” this post here and this article of D. Kleinbock and G. Margulis for a nice introductions to this circle of ideas), and it gives the following outcome when applied in our current setting.

Theorem 5 If {N_H} is non-degenerate, then there exists a constant {p} (depending only on {N_H}) such that, after a change of variables, the set

\displaystyle \{|c|<\kappa^{1/p}:\Omega(c)\in DC(\kappa, \tau_0)\}

has positive Lebesgue measure and its density at {0} converges to {1} as {\kappa\rightarrow 0}.

In particular, in the case {N_H} is not degenerate, this result (combined with our previous discussion) permits to recover a KAM-like theorem, i.e., it gives plenty (positive measure set) of KAM torii near the initial one.

On the other hand, in the general case (i.e., if we allow {N_H} to be degenerate), Theorem 3 has the following consequence

Corollary 6 A KAM torus for a real-analytic system is never isolated.

Evidently, even though this corollary allows to consider a general {H\in\mathcal{C}(\omega_0)}, it is “only” a partial answer to M. Herman’s question (as in the degenerate case there is not Lebesgue measure estimate).

In any case, the main difficulty one should face to completely answer the M. Herman’s question is clear: one should study the map {c\mapsto \Omega(c)} and understand how frequently it passes through Diophantine frequencies.

Concluding his talk, H. Eliasson pointed that real-analyticity is crucial and this discussion breaks down in the {C^{\infty}} category. More precisely, it is possible to construct an example with the features described in the next (final) section.

6. A {C^{\infty}} example

Consider {\omega_0\in\mathbb{R}^4} Diophantine and write {\omega_0=(\omega_1,\dots,\omega_4)}, {r=(r_1,\dots,r_4)}, and

\displaystyle h_0(r)=\langle\omega_0, r \rangle+f_1(r_4)r_1+f_2(r_4)r_2+f_3(r_4)r_3

The first 3 components of the gradient of {h_0} are

\displaystyle \nabla h_0(r) := (\Omega_1(r),\Omega_2(r),\Omega_3(r),\ast)

\displaystyle = (\omega_1+f_1(r_4), \omega_2+f_2(r_4), \omega_3+f_3(r_4),\ast)

Proposition 7 There are {C^{\infty}}-functions {f_1, f_2, f_3} with {f_j(0)=0}, {f_j} is {C^{\infty}} flat at {0} for all {j=1, 2, 3} such that for all {r_4\neq 0}, there are at least two entries of the vector

\displaystyle (\Omega_1(r), \Omega_2(r), \Omega_3(r))

which are Liouville.

In particular, using these functions, H. Eliasson, B. Fayad, and R. Krikorian are able to construct {C^{\infty}}-perturbations {H=h_0(r)+h_1(\phi,r)} of {h_0(r)} with {h_1(\phi,r)} {C^{\infty}}-flat at {r_4=0} such that the Hamiltonian flow {\varphi_t^H} associated to {H} satisfies

\displaystyle \lim\limits_{t\rightarrow\pm\infty}\|\varphi_t^H(\phi, r)\|=+\infty

for all {(\phi,r)} with {r_4\neq 0}.

Of course, this last property of divergence of orbits with initial data {(\phi,r)} with {r_4\neq0} forbids the existence of KAM torii containing such initial data. But, this is not a contradiction to a statement like Theorem 3 (or Corollary 6) because {r_4=0} contains KAM torii.

Remark 2 Apparently, M. Herman was aware of the existence of such examples (as he comments on his lecture notes for his ICM’98 talk), but H. Eliasson (and other people close to him) don’t know precisely how he intended to construct them because of his death in 2000.


Responses

  1. Edit (September 1, 2012): As it was kindly pointed out to me by Jacques Féjoz, the Kolmogorov’s non-degeneracy condition in the KAM theorem was incorrectly stated as \int_{\mathbb{T}^d} \det \partial_r^2 H(\phi, r=0) d\phi\neq 0. In fact, the correct condition (already incorporated in the text) is \det\int_{\mathbb{T}^d} \partial_r^2 H(\phi, r=0) d\phi\neq 0. (Thanks Jacques!)


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