Author
Jean-Pierre Marco
Other affiliations: University of Paris
Bio: Jean-Pierre Marco is an academic researcher from Pierre-and-Marie-Curie University. The author has contributed to research in topics: Integrable system & Hamiltonian system. The author has an hindex of 7, co-authored 17 publications receiving 120 citations. Previous affiliations of Jean-Pierre Marco include University of Paris.
Papers
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TL;DR: In this paper, the authors improved previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems and provided a sharper lower bound on the time of Arnold diffusion which they believe to be optimal.
Abstract: In this paper, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper lower bound on the time of Arnold diffusion which we believe to be optimal.
26 citations
TL;DR: In this article, the authors improved previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems and provided a sharper upper bound on the speed of Arnold diffusion which they believe to be optimal.
Abstract: In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion which we believe to be optimal.
20 citations
TL;DR: In this article, the authors studied the diagonal action of the unitary group U(n) on triples of Lagrangian subspaces of C n and gave a complete description of the orbit space.
18 citations
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TL;DR: It is proved that, under conditions on the critical level of the Bott first integral and dynamical conditions on an enregy level where it admits a first integral which is nondegenerate in the Bott sense, the weak polynomial entropy belongs to {0,1,2}.
Abstract: In this paper, we study the entropy of a Hamiltonian flow in restriction to an enregy level where it admits a first integral which is nondegenerate in the Bott sense. It is easy to see that for such a flow, the topological entropy vanishes. We focus on the polynomial and the weak polynomial entropies. We prove that, under conditions on the critical level of the Bott first integral and dynamical conditions on the hamiltonian function, the weak polynomial entropy belongs to {0,1} and the polynomial entropy belongs to {0,1,2}.
11 citations
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TL;DR: In this article, the authors introduce two numerical conjugacy invariants for dynamical systems, the complexity and weak complexity indices, which are well suited for the study of completely integrable Hamiltonian systems.
Abstract: We introduce two numerical conjugacy invariants for dynamical systems -- the
complexity and weak complexity indices -- which are well-suited for the study
of "completely integrable" Hamiltonian systems These invariants can be seen as
"slow entropies", they describe the polynomial growth rate of the number of
balls (for the usual "dynamical" distances) of coverings of the ambient space
We then define a new class of integrable systems, which we call decomposable
systems, for which one can prove that the weak complexity index is smaller than
the number of degrees of freedom Hamiltonian systems integrable by means of
non-degenerate integrals (in Eliasson-Williamson sense), subjected to natural
additional assumptions, are the main examples of decomposable systems We
finally give explicit examples of computation of the complexity index, for
Morse Hamiltonian systems on surfaces and for two-dimensional gradient systems
10 citations
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1,620 citations
01 Apr 1995
TL;DR: In this paper, the authors focus on identifying important specific properties associated with the asymptotic behavior of smooth dynamical systems, including growth of the numbers of orbits of various kinds and complexity of orbit families, types of recurrence, and statistical behavior of orbits.
Abstract: In this chapter we will embark upon the task of systematically identifying important specific phenomena associated with the asymptotic behavior of smooth dynamical systems We will build upon the results of our survey of specific examples in Chapter 1 as well as on the insights gained from the general structural approach outlined and illustrated in Chapter 2 Most of the properties discussed in the present chapter are in fact topological invariants and can be defined for broad classes of topological dynamical systems, including symbolic ones The predominance of topological invariants fits well with the picture that emerges from the considerations of Sections 21, 23, 24, and 26 The considerations of the previous chapter make it very plausible that smooth dynamical systems are virtually never differentiably stable and can only rarely be classified locally up to smooth conjugacy In contrast, structural and the related topological stability seem to be fairly widespread phenomena We will consider three broad classes of asymptotic invariants: (i) growth of the numbers of orbits of various kinds and of the complexity of orbit families, (ii) types of recurrence, and (iii) asymptotic distribution and statistical behavior of orbits The first two classes are of a purely topological nature; they are discussed in the present chapter The last class is naturally related to ergodic theory and hence we will provide an introduction to key aspects of that subject This will require some space so we put that material into a separate chapter The two chapters are intimately connected
572 citations
TL;DR: In this paper, the authors describe old and new motivations to study symplectic embedding problems and discuss a few of the many old and the many new results on the problem of embedding.
Abstract: We describe old and new motivations to study symplectic embedding problems, and we discuss a few of the many old and the many new results on symplectic embeddings.
51 citations
TL;DR: For a convex, real analytic, e-close to integrable Hamiltonian system with n ≥ 5 degrees of freedom, the authors constructed an orbit exhibiting Arnold diffusion with the diffusion time bounded by
Abstract: For a convex, real analytic, e-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded by $\exp(C\epsilon^{-\frac{1}{2(n-2)}})$
. This upper bound of the diffusion time almost matches the lower bound of order $\exp(\epsilon ^{-\frac{1}{2(n-1)}})$
predicted by the Nekhoroshev-type stability results. Our method is based on the variational approach of Bessi and Mather, and includes a new construction on the space of frequencies.
50 citations
TL;DR: In this article, it was shown that, for 0 e ≪ 1, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables I perform rather arbitrary excursions in a domain of size O ( 1 ).
36 citations