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Normal forms of analytic perturbations of quasihomogeneous vector fields: Rigidity, invariant analytic sets and exponentially small approximation

Eric Lombardi, Laurent Stolovitch (2010)

Annales scientifiques de l'École Normale Supérieure

In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of ℂ n , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. We study the normal form problem relatively to S . We give a condition on S that ensures that there...

Normal forms of invariant vector fields under a finite group action.

Federico Sánchez-Bringas (1993)

Publicacions Matemàtiques

Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Patrick Bonckaert, Freek Verstringe (2012)

Annales de l’institut Fourier

We explore the convergence/divergence of the normal form for a singularity of a vector field on ℂ n with nilpotent linear part. We show that a Gevrey- α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey- 1 + α type with the use of a Gevrey- 1 + α transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.

Normal points for generic hyperbolic maps

Mark Pollicott (2009)

Fundamenta Mathematicae

We consider families of hyperbolic maps and describe conditions for a fixed reference point to have its orbit evenly distributed for maps corresponding to generic parameter values.

Normalization of bundle holomorphic contractions and applications to dynamics

François Berteloot, Christophe Dupont, Laura Molino (2008)

Annales de l’institut Fourier

We establish a Poincaré-Dulac theorem for sequences ( G n ) n ∈ ℤ of holomorphic contractions whose differentials d 0 G n split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of ℂ ℙ k . In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...

Normalization of Poincaré singularities via variation of constants.

Timoteo Carletti, Alessandro Margheri, Massimo Villarin (2005)

Publicacions Matemàtiques

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on ( X ̃ , X ̃ , m ) . Consider the associated unitary operators on L ² ( X ̃ , X ̃ , m ) given by U ̃ S f = √ ( d ( m ∘ S ) / d m ) · ( f ∘ S ) and φ · U ̃ S , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

Notes on symplectic non-squeezing of the KdV flow

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao (2005)

Journées Équations aux dérivées partielles

We prove two finite dimensional approximation results and a symplectic non-squeezing property for the Korteweg-de Vries (KdV) flow on the circle 𝕋 . The nonsqueezing result relies on the aforementioned approximations and the finite-dimensional nonsqueezing theorem of Gromov [14]. Unlike the work of Kuksin [22] which initiated the investigation of non-squeezing results for infinite dimensional Hamiltonian systems, the nonsqueezing argument here does not construct a capacity directly. In this way our...

Nouvelle preuve d’un théorème de Yuan et Hunt

Thierry Bousch (2008)

Bulletin de la Société Mathématique de France

Un théorème de Guo-Cheng Yuan Brian R. Hunt affirme que, pour μ mesure de probabilité invariante d’un système dynamique hyperbolique T : X → X , les fonctions lipschitziennes X → ℝ pour lesquelles μ est minimisante ont un intérieur non vide (en topologie de Lipschitz) si et seulement si μ est une orbite périodique de T . Je donnerai une nouvelle preuve de ce théorème, ou plutôt d’un énoncé essentiellement équivalent. Je discuterai aussi de la stabilité des orbites périodiques minimisantes de grande période.

Numerical Approximations of the Dynamical System Generated by Burgers’ Equation with Neumann–Dirichlet Boundary Conditions

Edward J. Allen, John A. Burns, David S. Gilliam (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers’ equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approximations of this...

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