Santiago Barbieri

Postdoctoral Researcher in Mathematics

Universitat de Barcelona

With respect to the stability of those Hamiltonian systems which are close to integrable, two kinds of results exist:

**The results of stability in infinite time obtained in the framework of Kolmogorov-Arnol'd-Moser (KAM) theory**, which are valid for generic integrable Hamiltonian systems under the effect of a small enough perturbation. These theorems provide Cantor sets of big measure where the solutions of the considered systems are quasi-periodic. However, for system depending on more than two degrees of freedom, there is little information about other trajectories and instability may occur in the complementary of such Cantor sets (a phenomenon known as Arnol'd diffusion).

**The results of effective stability obtained in the framework of Nekhoroshev theory,**

In the same setting, N.N. Nekhoroshev proved in 1977 that, if the unperturbed Hamiltonian is Gevrey (Hölder) and steep, i.e. if its gradient satisfies a generic transversality condition, all solutions of the considered system are stable over a time which is exponentially (polynomially) long w.r.t. the inverse of the size of the perturbation. In particular all quasi-convex functions are steep.

Although in the last decades KAM Theory has been studied in the case of systems depending on an infinite number of degrees of freedom the same cannot be said of Nekhoroshev’s estimates, for which few generalizations exist. In particular, in the case of lattices - that is, of systems in R^Z - results are known only when the initial energy is localized in a finite number of modes. This kind of investigation is particularly interesting as lattices are used in the description of various physical systems, like crystals in solid state physics or the Fermi-Pasta-Ulam model. The difficulty of extending results of effective stability to this setting is mainly due to the fact that, in the finite dimensional case, Nekhoroshev’s estimates depend on the number of d.o.f. in such a way that they break down in the thermodynamical limit. However, Bourgain has recently introduced a version of the Diophantine non-resonant condition which is independent of the dimension. Since, in the finite-dimensional case, the proof of Nekhoroshev's estimates relies essentially on the existence of a suitable covering of the phase space with "resonant blocks" which are constructed starting from the classical Diophantine condition, the goal here is to build an infinite-dimensional equivalent of this patchwork starting from Bourgain's improved Diophantine condition. The choice of a "simple"perturbation containing only short range interactions should also be helpful in this kind

of investigation, as it was in the case in which Arnol’d’s diffusion on lattices was considered.

Arnold diffusion is a phenomenon of instability that may occur in nearly-integrable hamiltonian systems, with more than two degrees of freedom, for initial conditions that belong to the complementary of the set of invariant tori given by KAM theory. The goal of this research project is to construct an example of diffusive trajectory for a steep non-convex system. Indeed, though Arnol'd diffusion has been widely studied in the convex case, no examples of diffusion are known in the steep non-convex case. Moreover, the sought example should be such that the time after which the instability occurs can be computed: this would prove useful in order to study the optimality of the times of stability yielded by Nekhoroshev theory.

The steepness property is a transversality condition on the gradient of a given function that is fundamental in Nekhoroshev Theory. Its original definition is quite involved, but one can prove (see Niederman, Annales de l'Institut Fourier, 2006) that an analytic function defined on a compact set is steep iff it has no critical points and any of its restrictions to any affine subspace has only an isolated set of critical points. Therefore, convex functions are steep, as they only have non-degenerated critical points. However, whereas convexity is just an open property, steepness turns out to be generic both in measure and topological sense in the set of Taylor polynomials of smooth functions. The proof of this fact is due to Nekhoroshev (1973) and exploits challenging techniques of real algebraic geometry and complex analysis. In particular, Nekhoroshev proofs particular cases of important results of real-algebraic geometry which were not known at the time he was writing, but that are now well known; moreover, his setting is adapted to the specific problem of the genericity of steepness, which makes the proof rather obscure. During my thesis, I have simplified Nekhoroshev's proof by making use of more modern tools of real-algebraic geometry. As a byproduct, I have rediscovered a result about Bernstein inequalities for algebraic functions that is hidden in Nekhoroshev's work (1973) and that was later proven in a different way by Roytwarf and Yondim in the Nineties. Moreover, this work has led me to discover new explicit sufficient conditions to ensure that a given function is steep, a fundamental step in order to apply Nekhoroshev estimates to physical systems.

Nekhoroshev theory insures stability of a nearly-integrable system over an exponentially long time (in the inverse of the size of the perturbation) only if the hamiltonian at hand is analytic or belongs to the Gevrey class. For lower regularities one can only obtain polynomially long times.

With my colleagues, I have worked on this topic by extending Nekhoroshev's estimates to the class of Hölder steep functions and obtained refined estimates for the parameters of stability. As a byproduct, we have conceived a new technique of proving stability that makes use of an improved theorem of analytic smoothing for Hölder functions.

The research for the best estimates of stability for nearly-integrable steep systems is important in order to understand the link to the study of Arnol'd diffusion in the steep case, a field which has not yet been explored.

The KAM theorem claims that the phase space of a non-degenerate nearly integrable Hamiltonian system is foliated by invariant tori up to a set whose measure is smaller than √ε, where ε is the size of the

perturbation. A conjecture of Arnold, Kozlov and Neishtadt says that for generic real-analytic Hamiltonian systems, this measure is actually smaller, i.e. proportional to ε. Luca Biasco and Luigi Chierchia have announced and proved this conjecture for real-analytic mechanical systems by studying the measure of the invariant tori close to single resonances, i.e. to those parts of the phase space where the frequencies of the unperturbed hamiltonian satisfy a single linear homogeneous equation with integer coefficients. Their proof holds for hamiltonian functions of the kind

H(I,x)=I^2/2+ε V(x)

where (I,x) ∈ R^n x T^n are the standard action-angle coordinates. The goal of this research project consists in extending this result to more general hamiltonian systems of the kind

H(I,x)=h(I)+ε V(I,x)

with h convex or steep.

Up to now, I have worked with L. Biasco on an extension of this result to the case

H(I,x)=I^2/2+ε V(I,x) .

Indeed, the simple fact to add a dependence of the potential on the action variables, adds up many more difficulties to the problem. More precisely, one of the key steps in the proof of the result for potentials depending only on the angle variables consists in proving that close to single resonances the hamiltonian can be generically conjugated to a "pendulum-like" system with one degree of freedom by the means of a symplectic transformation. Proving the same result in the case in which V depends also on the actions requires additional considerations of quantitative Morse-Sard theory.

It is known that steepness is only a sufficient condition for exponential stability, as there exists a wider prevalent set of functions, the so-called diophantine steep class, whose elements are exponentially stable under the effect of any sufficiently small perturbation. Differently from the steep case, where refined estimates for the time of stability exist, the known estimates of stability in the diophantine-steep case are not sharp since they have been obtained by making use of a strategy of proof which is different from Nekhoroshev's original one. The aim of this axis of research is therefore to find refined estimates for the time of stability by applying Nekhoroshev's original technique to the class of diophantine steep functions. This result would be of great interest since it would be a {\bf first step in order to understand up to which point exponential stability holds for systems that are "intermediate" between steep ones and non-steep non-stable ones.}

I have worked on some applications of Nekhoroshev theory to the three-body problem (both in the planetary and in the restricted case). One of the main issues when studying this kind situations is that the size of the perturbation - given by the ratio of the largest among the masses of the two planets to the mass of the central attractor - might not be small enough for the mathematical theorem to apply in reality. In particular, I have investigated how the mathematical theorems cease to apply when the size of the perturbation reaches values which are too high.

This axis of research is devoted to answering a question by M. Bialy on the special class of "coin billiards", i.e. dynamical systems described by a twist map of the annulus which is the composition of the standard billiard map for a convex subset A of R^2 with the geodesic flow associated to the surface of the cylinder of fixed height d associated to A. Bialy’s question amounts to asking whether invariant KAM curves exist in this kind of systems, and whether the system is integrable when A is the ellipse. Coin billiards are interesting as their flow is equivalent to a non-smooth geodesic flow on a two-sphere. Up to now, with my collaborator, we have been able to prove the existence of KAM curves at fixed distance from the boundary in case A is close to a circle or the height d of the cylinder is very small. We also have numerical evidence that the elliptic coin billiard is not integrable, and currently seeking for an analytic proof. We would also like to show the existence of Aubry-Mather sets near the boundary.

(with J.-P. Marco and T. Dias Oliveira Silva)

The main idea behind this axis is the fact that, by Khovanski's theory of fewnomials, given a real algebraic variety (or even a transcendental one, in some cases), its topological complexity depends on the combinatorial complexity of the equations that define it, i.e. on the number of monomials appearing in the equations that determine the variety (the simplest example of this fact is Descartes’ rule on the number of positive and negative solutions of a real polynomial equation of one variable). In particular, explicit refined bounds on the Betti numbers of algebraic varieties and semi-algebraic sets in terms of the combinatorial complexity of the equations that determine them are given in a recent paper by Basu and Rizzie. The first goal of this axis of research, then, would be to give bounds on the Betti numbers of the semi-algebraic set containing the Taylor polynomials of non-steep functions at a given order, as the monomials appearing in the equalities and inequalities determining this set can be controlled explicitly with the help of algorithms of real-algebraic geometry that only involve linear operations (this last aspect is important, as it simplifies computations dramatically). Then, as a second step, one could try to see whether the topological properties of this set are related to dynamical properties of the Hamiltonian functions whose r-th order Taylor polynomials lie in it.