arXiv subjects

Santiago Barbieri

Publications and source records attributed to Santiago Barbieri.

10 recordsLinked to original sources

arXiv2026

Nekhoroshev Theorem for time quasiperiodic perturbations of P-Steep systems

We prove a Nekhoroshev type result for a time quasiperiodic perturbation of an integrable Hamiltonian system. More precisely, we assume that the integrable part is analytic and fulfills a generic nondegeneracy condition introduced by Nekhoroshev and called P-Steepness. We add a small perturbation which depends in a quasiperiodic way on time (with Diophantine frequency) and prove that -- for times exponentially long with the inverse of the size $\varepsilon$ of the perturbation -- the actions of the unperturbed system remain approximately constant. The proof is based on an extension to the time dependent case of the proof {of classical Nekhoroshev's theorem} given by Guzzo, Chierchia and Benettin, which however requires new ideas in order to deal with the more complex geometry of resonances of the time dependent case.

math.DS↗

arXiv2025

Singular KAM theory for convex Hamiltonian systems

In this note, we briefly discuss how singular KAM Theory - which was worked out in a previous work by L.B. and L.C. for the mechanical case $\frac12 |y|^2+\varepsilon f(x)$ - can be extended to convex real analytic nearly integrable Hamiltonian systems with Hamiltonian in action-angle variables given by $h(y)+\varepsilon f(x)$ with $h$ convex and generic $f$.

math.DS↗

arXiv2025

Existence and Nonexistence of Invariant Curves of Coin Billiards

In this paper we consider the coin billiard introduced by M. Bialy. It is a modification of the classical billiard, obtained as the return map of a nonsmooth geodesic flow on a cylinder that has homeomorphic copies of a classical billiard on the top and on the bottom (a coin). The return dynamics is described by a map $T$ of the annulus $\mathbb A = \mathbb T \times (0,π)$. We prove the following three main theorems: in two different scenarios (when the height of the coin is small, or when the coin is near-circular) there is a family of KAM curves close to, but not accumulating on, the boundary $\partial \mathbb A$; for any noncircular coin, if the height of the coin is sufficiently large, there is a neighbourhood of $\partial \mathbb A$ through which there passes no invariant essential curve; and the only coin billiard for which the phase space $\mathbb A$ is foliated by essential invariant curves is the circular one. These results provide partial answers to questions of Bialy. Finally, we describe the results of some numerical experiments on the elliptical coin billiard.

math.DS↗

arXiv2024

Semi-algebraic Geometry and generic Hamiltonian stability

The steepness property is a local geometric transversality condition on the gradient of a $C^2$-function which is fundamental in order to ensure the stability of sufficiently-regular nearly-integrable Hamiltonian systems over long timespans. Steep functions were originally introduced by Nekhoroshev, who also proved their genericity. Namely, given a pair of positive integers $r,n$, with $r$ high enough, and a point $I_0\in \mathbb{R}^n$, the Taylor polynomials of those $C^{2r-1}$ functions which are not steep around $I_0$ are contained in a semi-algebraic set of positive codimension in the space of polynomials of $n$ variables and degree bounded by $r$. The demonstration of this result was originally published in 1973 and has been hardly studied ever since, probably due to the fact that it involves no arguments of dynamical systems: it makes use of quantitative reasonings of real-algebraic geometry and complex analysis. The aim of the present work is two-fold. In the first part, the original proof of the genericity of steepness is rewritten by making use of modern tools of real-algebraic geometry: this allows to clarify the original reasonings, that were obscure or sketchy in many parts. In particular, Yomdin's Lemma on the analytic reparametrization of semi-algebraic sets, together with non trivial estimates on the codimension of certain algebraic varieties, turn out to be the fundamental ingredients to prove the genericity of steepness. The second part of this work is entirely new and is devoted to the formulation of explicit algebraic criteria to check steepness of any given sufficiently regular function, which constitutes a very important result for applications, as the original definition of steepness is not constructive. These criteria involve both the derivatives of the studied function up to any given order and external real parameters that, generically, belong to compact sets.

math.AG↗

arXiv2024

Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians

We prove that the solutions of Hölder-differentiable Hamiltonian systems, associated to initial conditions in a small ball of radius $ρ>0$ around a Lagrangian, $(γ,τ)-$Diophantine, quasi-periodic torus, are stable over a time $t^{\text{stab}}\simeq 1/(|ρ|^{1+\frac{\ell-1}{τ+1}}|\ln ρ|^{\ell-1})$, where $\ell>2d+1, \ell \in \mathbb R$, is the regularity, and $d$ is the number of degrees of freedom. In the finitely differentiable case (for integer $\ell$), this result improves the previously known effective stability bounds around Diophantine tori. Moreover, by a previous work based on the Anosov-Katok construction, it is known that for any $\varepsilon>0$ there exists a $C^\ell$-Hamiltonian, with $ \ell\ge 3$, admitting a sequence of solutions starting at distance $ρ_n \to 0$ from a $(γ,τ)$-Diophantine torus that diffuse in a time of order $t^{\text{diff}}_n\simeq 1/(|ρ_n|^{1+\frac{\ell-1}{τ+1}+\varepsilon})$. Therefore the stability estimates that we show are optimal up to an arbitrarily small polynomial correction.

math.DS↗

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.

math.DS↗

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 $k$.

math.DS↗

arXiv2022

Bernstein-Remez inequality for Nash functions: A complex analytic approach

Consider an open, bounded set $Ω\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset Ω$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline Ω$, and whose graph satisfies $S(z,f(z))=0$ for some polynomial $S\in\mathbb{C}[z,w]$ of degree at most $k$ (hence $f$ is an algebraic function), the quantity $\max_{\overlineΩ}|f|/\max_{\mathcal{K}}|f|$ is bounded by a constant that only depends on $k$, $Ω$, $\mathcal{K}$ but not on $f$ (estimates of this kind are called Bernstein-Remez inequalities). This result has been demonstrated by Roytwarf and Yomdin in case $\mathcal{K}$ is a real interval, and later by Yomdin for a discrete set $\mathcal{K}$ of sufficiently high cardinality, by using arguments of real-algebraic and analytic geometry. Here we present and extend a proof due to Nekhoroshev on the existence of a uniform Bernstein-Remez inequality for algebraic functions, which relies on classical theorems of complex analysis. Nekhoroshev's work remained unstudied despite its important consequences in Hamiltonian dynamics and is here presented and extended in a self-contained and pedagogical way, while the original reasonings were rather sketchy.

math.AG↗

arXiv2020

On the algebraic properties of exponentially stable integrable hamiltonian systems

Steepness is a geometric property which, together with complex-analyticity, is needed in order to insure stability of a near-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoro-shev, we construct sufficient algebraic conditions for steepness for a given function that involve algebraic equations on its derivatives up to order five. The underlying analysis suggests some interesting considerations on the gener-icity of steepness and represents a first step towards the construction of sufficient conditions for steepness involving the derivatives of the studied function up to an arbitrary order.

math.DS↗

arXiv2018

Sharp Nekhoroshev estimates for the three body problem around periodic orbits

We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.

math-ph↗