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).
- 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 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 (where is real-analytic of and whose first derivative at vanishes) obtained by perturbation of (the completely integrable Hamiltonian) . Here, are “angle-variables”, are “action-variables”, , and the frequency is assumed to be Diophantine, i.e.,
for some , . For sake of simplicity, we will denote this class of Hamiltonian’s by .
The Hamiltonian system associated to is the dynamical system (flow) defined by:
For , we see that is a real-analytic flow-invariant Lagrangian torus with flow . 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 .
Concerning the existence of these objects (KAM torii), we have the classical and celebrated KAM theorem:
Theorem (KAM’60, Pöschel’82). Let . If Kolmogorov’s non-degeneracy condition
holds, then the set of KAM torii has positive measure and it has density at
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).
Is the KAM theorem above true for all (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 , say . 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 by getting rid of the dependence on :
Proposition 1 Let . Then, there are
- analytic on , formal in , and symplectic (with respect to );
such that . Moreover,
- is unique and it is called the Birkhoff normal form (BNF) of
- is unique modulo “normalization” and it is called Birkhoff normalization of .
By the seminal works of H. Poincaré and C. Siegel, It is known that is “typically” divergent. For instance, H. Poincaré showed that diverges for topologically typical (generic) 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 ).
On the other hand, very little is known about . For instance, the following questions are open:
- Can be divergent?
- Can be convergent when is divergent?
Among the few facts known about , one has the following important result:
Theorem (R. Perez-Marco’03). If is divergent for some , then is divergent for “typical” .
In a similar vein, one can consider the set
and one can ask whether this set is “large” or “small”. Recently, A. Bounemoura proved that:
to move points in the -neighborhood of outside the -neighborhood of it. Here, 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 (e.g., ), then and are convergent.
Motivated by this result, one introduces the following notion:
Definition 2 We say that is -degenerate (for ) if and only if there are linearly independent frequency vectors such that
Remark 1 The assumption of H. Russmann’s theorem correspond to (i.e., -degenerate BNF), while it is not hard to check that Kolmogorov’s non-degeneracy condition in KAM theorem implies (i.e., -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):
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 such that:
- (here is the ball of radius around )
- for some (recall that when ).
real-analytic in for close to , and
a real-analytic function in and a -function in such that
where is -flat (i.e., and all its derivatives vanish) at .
In other words, this result says that after “adjusting” the Hamiltonian with the “counter-term” , we can perform an exact symplectic change of variables to get a nice “normal form” at sufficiently Diophantine frequencies .
Here, the “range” of the change of variables depends on (as well as the big 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
Furthermore, if (i.e., “passes” by Diophantine frequencies), then the –jet of the term above vanishes (i.e., is -flat at ) and this allows us to deduce the presence of KAM torii (by setting ).
So, this leads us to study . We know that
- (a) the –jet of is (by uniqueness of BNF);
- (b) .
Now, when the is -degenerate, we can find linearly independent unitary vectors such that for all , .
In particular, by item (a) above, we get that
for all .
Therefore, by item (b), we conclude that, for all ,
where is a function of the partial derivatives of and determined by Leibnitz formula.
In our situation, we know (see above) that for all , so that it is not hard to deduce that , and, a fortiori,
for all .
However, since the function is real-analytic (see the statement of Theorem 4), we conclude that
for all close to , that is,
for all close to , and, as we mentioned above, this gives a real-analytic submanifold of dimension close to and foliated by KAM torii with frequency vector . Thus, this completes the sketch of proof of Theorem 3.
5. Back to M. Herman’s problem
Actually, in the case is non-degenerate, one can use the analyticity (and non-degeneration) properties of to show that, since “starts” at a Diophantine frequency (in the sense that ), 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 is non-degenerate, then there exists a constant (depending only on ) such that, after a change of variables, the set
has positive Lebesgue measure and its density at converges to as .
In particular, in the case 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 to be degenerate), Theorem 3 has the following consequence
Evidently, even though this corollary allows to consider a general , 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 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 category. More precisely, it is possible to construct an example with the features described in the next (final) section.
6. A example
Consider Diophantine and write , , and
The first 3 components of the gradient of are
Proposition 7 There are -functions with , is flat at for all such that for all , there are at least two entries of the vector
which are Liouville.
In particular, using these functions, H. Eliasson, B. Fayad, and R. Krikorian are able to construct -perturbations of with -flat at such that the Hamiltonian flow associated to satisfies
for all with .
Of course, this last property of divergence of orbits with initial data with 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 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.