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Jean-Pierre Marco

Publications and source records attributed to Jean-Pierre Marco.

13 recordsLinked to original sources
arXiv2022

Flexibility and analytic smoothing in averaging theory

Using a new strategy, we extend the classical Nekhoroshev's estimates to the case of Hölder regular steep near-integrable hamiltonian systems, the stability times being polynomially long in the inverse of the size of the perturbation. We prove that the stability exponents can be taken to be $(\ell-1)/(2nα_1...α_{n-2})$ for the time of stability and $1/(2nα_1...α_{n-1})$ for the radius of stability, $\ell >n+1$ being the regularity and the $α_i$'s being the indices of steepness. Our strategy consists in deriving a perturbation theory which exploits a sharp analytic smoothing theorem to approximate any Hölder function by an analytic one. In addition, an appropriate choice of the free parameters in the problem enables us to have a first grasp on the relation connecting the time and radius of stability to the threshold that the size of the perturbation must satisfy in order for the theorem to apply. Particular attention is payed to a geometric presentation of the construction of the so-called "resonant blocks", in order to shed a definitive light on the nature of the steepness condition. We also investigate the convex setting, using a similar approach.

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arXiv2022

Analytic Smoothing and Nekhoroshev estimates for Hölder steep Hamiltonians

In this paper we prove the first result of Nekhoroshev stability for steep Hamiltonians in Hölder class. Our new approach combines the classical theory of normal forms in analytic category with an improved smoothing procedure to approximate an Hölder Hamiltonian with an analytic one. It is only for the sake of clarity that we consider the (difficult) case of Hölder perturbations of an analytic integrable Hamiltonian, but our method is flexible enough to work in many other functional classes, including the Gevrey one. The stability exponents can be taken to be $(\ell-1)/(2n{\mathbfα}_1...{\mathbfα}_{n-2})+1/2$ for the time of stability and $1/(2n{\mathbfα}_1...{\mathbfα}_{n-1})$ for the radius of stability, $n$ being the dimension, $\ell >n+1$ being the regularity and the ${\mathbfα}_i$'s being the indices of steepness. Crucial to obtain the exponents above is a new non-standard estimate on the Fourier norm of the smoothed function. As a byproduct we improve the stability exponents in the $C^k$ class, with integer $k$.

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arXiv2022

Diffusion along chains of normally hyperbolic cylinders

We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are $3$-dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders -- i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition -- and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders -- i.e., certain compatibility conditions between the tilt map and the homoclinic maps attached to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems.

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arXiv2021

Computer Assisted Proof of Drift Orbits Along Normally Hyperbolic Manifolds

Normally hyperbolic invariant manifolds theory provides an efficient tool for proving diffusion in dynamical systems. In this paper we develop a methodology for computer assisted proofs of diffusion in a-priori chaotic systems based on this approach. We devise a method, which allows us to validate the needed conditions in a finite number of steps, which can be performed by a computer by means of rigorous-interval-arithmetic computations. We apply our method to the generalized standard map, obtaining diffusion over an explicit range of actions.

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arXiv2016

Chains of compact cylinders for cusp-generic nearly integrable convex systems on $\mathbb{A}^3$

This paper is the first of a series of three dedicated to a proof of the Arnold diffusion conjecture for perturbations of {convex} integrable Hamiltonian systems on $\mathbb{A}^3=\mathbb{T}^3\times \mathbb{R}^3$. We consider systems of the form $H(θ,r)=h(r)+f(θ,r)$, where $h$ is a $C^κ$ strictly convex and superlinear function on $\mathbb{R}^3$ and $f\in C^κ(\mathbb{A}^3)$, $κ\geq2$. Given $e>\textrm{Min}\,h$ and a finite family of arbitrary open sets $O_i$ in $\mathbb{R}^3$ intersecting $h^{-1}(e)$, a diffusion orbit associated with these data is an orbit of $H$ which intersects each open set $\widehat O_i=\mathbb{T}^3\times O_i\subset\mathbb{A}^3$. The first main result of this paper (Theorem I) states the existence (under cusp-generic conditions on $f$ in Mather's terminology) of "chains of compact and normally hyperbolic invariant $3$-dimensional cylinders" intersecting each $\widehat O_i$. Diffusion orbits drifting along these chains are then proved to exist in subsequent papers. The second main result (Theorem II) consists in a precise description of the hyperbolic features of classical systems (sum of a quadratic kinetic energy and a potential) on $\mathbb{A}^2=\mathbb{T}^2\times\mathbb{R}^2$, which is a crucial step to prove Theorem I.

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arXiv2016

Arnold diffusion for cusp-generic nearly integrable convex systems on ${\mathbb A}^3$

We prove the existence of "Arnold diffusion orbits" in cusp-generic nearly integrable a priori stable systems on ${\mathbb A}^3$. The result relies on the cusp-generic existence of chains in nearly integrable a priori stable systems, proved in a previous paper, together with the existence of diffusion orbits along such chains. The latter result was obtained in collaboration with M. Gidea.

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arXiv2015

Measure and capacity of wandering domains in Gevrey near-integrable exact symplectic systems

A wandering domain for a diffeomorphism is an open connected set whose iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact symplectic structure. An integrable diffeomorphism Φ^h, i.e. the time-one map of a Hamiltonian h which depends only on the action variables, has no nonempty wandering domains. The aim of this paper is to estimate the size (measure and Gromov capacity) of wandering domains in the case of an exact symplectic perturbation of Φ^h , in the analytic or Gevrey category. Upper estimates are related to Nekhoroshev theory, lower estimates are related to examples of Arnold diffusion. This is a contribution to the "quantitative Hamiltonian perturbation theory" initiated in previous works on the optimality of long term stability estimates and diffusion times; our emphasis here is on discrete systems because this is the natural setting to study wandering domains.

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arXiv2014

Examples of nearly integrable systems on $\mathbb{A}^3$ with asymptotically dense projected orbits

Given an integer $κ\geq2$, we introduce a class of nearly integrable systems on $\mathbb{A}^3$, of the form $$ H_n(θ,r)=\frac12 \Vert r\Vert ^2+\tfrac{1}{n} U(θ_2,θ_3)+f_n(θ,r) $$ where $U\in C^κ(\mathbb{T}^2)$ is a generic potential function and $f_n$ a $C^{κ-1}$ additional perturbation such that $\Vert f_n\Vert_{C^{κ-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\frac12\Vert r\Vert ^2$. Let $Π:\mathbb{A}^3\to\mathbb{R}^3$ be the canonical projection. We prove that for each $δ>0$, there exists $n_0$ such that for $n\geq n_0$, the system $H_n$ admits an orbit $Γ_n$ at energy $\frac12$ whose projection $Π(Γ_n)$ is $δ$-dense in $Π(H_n^{-1}(\tfrac{1}{2}))$, in the sense that the $δ$-neighborhood of $Π(Γ_n)$ in $\mathbb{R}^3$ covers $Π(H_n^{-1}(\frac{1}{2}))$.

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arXiv2012

Polynomial entropies for Bott nondegenerate Hamiltonian systems

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}.

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arXiv2009

Dynamical complexity and symplectic integrability

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.

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