arXiv subjects

Mauricio Garay

Publications and source records attributed to Mauricio Garay.

At least 19 recordsLinked to original sources

arXiv2026

The Hamiltonian Normal Form

An important step in the proof of the Herman invariant tori conjecture was the introduction of a normal form with poles along the resonance loci, replacing the Birkhoff normal form, which we call the Hamiltonian normal form. This paper is extracted from previous versions (arXiv:1206.1245, arXiv:1909.06053) and aims to present this Hamiltonian normal form in its simplest form. It is expected that, not only theoretically but also in numerical computations, it will provide better approximations than the standard Birkhoff normal form.

math.DS↗

arXiv2025

Fixed point theorems for small divisors problems

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems.In practice, our theorems show that Bruno conditions are sufficient to ensure the existence of a positive measure set of invariant tori.

math.DS↗

arXiv2024

A new quasi-analytic class

Spaces of quasi-analytic classes are defined by the existence and uniqueness of Taylor expansions, which are not necessarily convergent. First examples were given by Borel in his theory of monogenic functions, a generalisation of holomorphic functions defined on locally closed sets. Denjoy and Carleman then gave simpler examples of quasi-analytic classes which are now widely known. Unfortunately, in most examples coming from mathematical physics and number theory, the power series are neither of Borel nor Denjoy-Carleman's classes. In this paper we introduce a quasi-analytic class which is relevant to perturbation theory and especially to KAM theory and dynamical systems. Our theorems also explain geometrically the divergence of most perturbative expansions by the presence of accumulation points of poles.

math.CV↗

arXiv2024

Non degenerate Birkhoff functions

It is known after the works of Mahler, Kleinbock, Margulis, Sprind\v{z}uk and others that very well approximated numbers on a manifold form a zero measure set, assuming non-degeneracy conditions. These non-degeneracy conditions are, in many applications, difficult to check. We propose here a setting in which these are automatic. Rather than functions, we consider correspondences which have a better behaviour.

math.DS↗

arXiv2023

A Category of Banach Space Functors

We introduce a sheaf theoretic viewpoint on functional analysis designed for infinite dimensional Lie group actions. We develop functional calculus for Banach valued functors and, in particular, prove the existence of an exponential map for a certain class of operators that generalise first order partial differential operators. This algebraic framework can be used in dynamical systems and KAM theory to provide normal forms and versal deformations. it is used in our papers on the subject such as the Herman conjecture paper, the versal deformation theorem for vector fields etc.

math.CV↗

arXiv2022

The Herman invariant tori conjecture

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/category/alg-geom/

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arXiv2020

Versal deformations of vector field singularities

When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant cylinders. In his thesis, Stolovitch established a new type of normal form and proved that for a simple resonance and under arithmetic conditions the cone is (the germ of) an analytic variety. In this paper, we prove a versal deformation theorem for analytic vector fields with an isolated singularity over Cantor sets. Our result implies that, under arithmetic conditions, the resonant cone is the degeneration of a set of invariant manifolds like in Pyartli's example. For the multi-Hopf bifurcation, that is for the $(-1,1)^d$-resonance, this implies the existence of vanishing tori carrying quasi-periodic motions generalising previous results of Chenciner and Li.

math.DS↗

arXiv2020

Quasi-periodic motions on symplectic tori

The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.

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arXiv2018

KAM THEORY. Part 3. Applications

We apply the general normal form theorems in Kolmogorov spaces to three classical cases: deformations of hypersurface singularities, normal forms of vector fields and invariant tori in Hamiltonian systems.

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arXiv2018

KAM Theory. Part II. Kolmogorov spaces

This is part II of our book on KAM theory. We start by defining functorial analysis and then switch to the particular case of Kolmogorov spaces. We develop functional calculus based on the notion of local operators. This allows to define the exponential and therefore relation between Lie algebra and Lie group actions in the infinite dimensional context. Then we introduce a notion of finite dimensional reduction and use it to prove a fixed point theorem for Kolmogorov spaces. We conclude by proving general normal theorems.

math.DS↗

arXiv2018

KAM Theory. Part I. Group actions and the KAM problem

This is part I of a book on KAM theory. We start from basic symplectic geometry, review Darboux-Weinstein theorems action angle coordinates and their global obstructions. Then we explain the content of Kolmogorov's invariant torus theorem and make it more general allowing discussion of arbitrary invariant Lagrangian varieties over general Poisson algebras. We include it into the general problem of normal forms and group actions. We explain the iteration method used by Kolmogorov by giving a finite dimensional analog. Part I explains in which context we apply the theory of Kolmogorov spaces which will form the core of Part II.

math.DS↗

arXiv2015

Théorie KAM

Lectures given on KAM theory at the University of Ouargla (Algeria). I present a functional analytic treatment of the subject which includes KAM theory into the general framework of deformations and singularity theory.

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arXiv2014

Sur le théorème KAM

In this short note, I explain how the non-degeneracy condition of the KAM can be bypassed. The first version of the note has been submitted for publication back in 2010 and this version in 2012.

math.DS↗

arXiv2014

Resurgent Deformation Quantisation

We construct a version of the complex Heisenberg algebra based on the idea of endless analytic continuation. In particular, we exhibit an integral formula for the product of resurgent operators with algebraic singularities. This algebra would be large enough to capture quantum effects that escape ordinary formal deformation quantisation.

math-ph↗

arXiv2013

The Herman conjecture

In the nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophantine critical point of a hamiltonian function. I show that KAM versal deformation theory solves positively this conjecture.

math.DS↗

arXiv2013

Degenerations of invariant lagrangian manifolds

We consider a pair (H,I) where I is an involutive ideal of a Poisson algebra and H lies in I. We show that if I defines a 2n-gon singularity then, under arithmetical conditions on H, any deformation of H can "integrated" as a deformation of (H,I).

math.DS↗

arXiv2013

An abstract KAM theorem

The KAM iterative scheme turns out to be effective in many problems arising in perturbation theory. I propose an abstract version of the KAM theorem to gather these different results.

math.DS↗

arXiv2013

Détermination finie sur un espace de Stein

Consider the ring of holomorphic function germs in $C^n$ and denote by $M$ the maximal ideal of this ring. For any a holomorphic function germ $f$ with an isolated critical point, the finite determinacy theorem (Mather-Tougeron) asserts that there exists some $k$, such that $f+g$ can be brought back to $f$, via a holomorphic change of variables, for any $g \in M^k$. In this paper, a generalisation of this theorem for functions defined in a neighbourhood of a Stein compact subset and for an arbitrary ideal is given.

math.AG↗