homoclinic orbits to invariant tori near a homoclinic orbit to center center saddle equilibrium we consider a perturbation of an integrable hamiltonian vector field with three degrees of freedom with a center center saddle equilibrium having a homoclinic orbit or loop with the help of a poincare map chosen based on the unperturbed homoclinic loop we study the homoclinic intersections between the stable and unstable manifolds associated to persistent hyperbolic kam tori on the center manifold near the equilibrium if the perturbation is such that the homoclinic loop is preserved i e the perturbation also has a homoclinic loop inherited from the unperturbed one we establish that in general the manifolds intersect along 8 12 or 16 transverse homoclinic orbits on the other hand in a more generic situation the loop is not preserved a condition for this fact is obtained by means of a melnikov like method the manifolds intersect along four transverse homoclinic orbits though a small neighborhood of the loop has to be excluded in a first approximation those homoclinic orbits can be detected as nondegenerate critical points of a melnikov potential defined on the 2 torus the number of homoclinic orbits is given by morse theory applied to the melnikov potential

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

degenerate lower dimensional tori in hamiltonian systems abstract we study the persistence of lower dimensional tori in hamiltonian systems of the form h x y z y 1 2 z m z e p x y z where x y z t n r n r 2 m e is a small parameter and m can be singular we show under a weak melnikov nonresonant condition and certain singularity removing conditions on the perturbation that the majority of unperturbed n tori can still survive from the small perturbation as an application we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable properly degenerate hamiltonian system for which neither the kolmogorov nor the g nondegenerate condition is satisfied

periodic and quasi periodic solutions of nonlinear wave equations via kam theory in this paper the nonlinear wave equation u u u xx v x u x t varepsilon u 3 x t 0 is studied it is shown that for a large class of potentials v x one can use kam methods to construct periodic and quasi periodic solutions in time for this equation

the spectrum of relativistic one electron atoms according to bethe and salpeter bethe and salpeter introduced a relativistic equation different from the bethe salpeter equation which describes relativistic multi particle systems here we will begin some basic work concerning its mathematical structure in particular we show self adjointness of the one particle operator which will be a consequence of a sharp sobolev type inequality yielding semi boundedness of the corresponding sesquilinear form moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum

the vlasov equation with infinite mass we show existence and uniqueness for the solutions to the vlasov equation for a positive smooth and short range potential in the case of unbounded mass

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

kam theory the legacy of kolmogorov s 1954 paper kolmogorov arnold moser or kam theory was developed for conservative dynamical systems that are nearly integrable integrable systems in their phase space usually contain lots of invariant tori and kam theory establishes persistence results for such tori which carry quasi periodic motions we sketch this theory which begins with kolmogorov s pioneering work

persistence of lower dimensional tori of general types in hamiltonian systems this work is a generalization to a result of j you 1999 we study the persistence of lower dimensional tori of general type in hamiltonian systems of general normal forms by introducing a modified linear kam iterative scheme to deal with small divisors we shall prove a persistence result under a melnikov type of non resonance condition which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori

a proof of the isoenergetic kam theorem from the ordinary one a proof is given of the isoenergetic kam theorem for hamiltonian systems using the ordinary kam theorem and a transversality argument

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

cantor spectrum for the almost mathieu operator in this paper we use results on reducibility localization and duality for the almost mathieu operator left h b phi x right n x n 1 x n 1 b cos left 2 pi n omega phi right x n on l 2 and its associated eigenvalue equation to deduce that for b 0 2 and diophantine the spectrum of the operator is a cantor subset of the real line this solves the so called ten martini problem for these values of b and moreover we prove that for b 0 small or large enough all spectral gaps predicted by the gap labelling theorem are open

singularities of normal forms and topology of orbits in area preserving maps the existence of analytic normal forms for an area preserving map with an elliptic fixed point is considered if the linear frequency is diophantine the complexified map can be analytically conjugated with an integrable map in a disc of the complex radial coordinate r excluding a sequence of sets whose measure decreases exponentially fast as the origin is approached the analyticity in the angle theta is just a strip the non analyticity domains correspond to the regions where the topology of the orbits changes since nonlinear resonances are present and the modulus of the residue of the leading poles provided by perturbation theory is the square of the width of the corresponding chains of islands this suggests a relation between clusters of singularities of the normalizing transformation and changes in the topology of the orbits

the screened poisson vlasov system with infinite charge and vanishing screening the vlasov equation may be used to model the dynamics of a collisionless plasma we consider a situation where the plasma has unbounded support and the charge density does not decay as x the electric field is computed from the screened poisson equation whose fundamental solution decays exponentially as x and hence gives a convergent integral despite the lack of decay of the charge density the limit as the screening is removed is studied in one space dimension it is shown that if some assumptions on the initial conditions hold then the limit in question exists but is not uniform and the problem satisfied by the limit is identified

a kam theorem for finitely differentiable hamiltonian systems abstract given l 2 2 d 4 we prove the persistence of a cantor family of kam tori of measure o e 1 2 l for any non degenerate nearly integrable hamiltonian system of class c l d t d where 1 is the diophantine power of the frequencies of the persistent kam tori and d r d is a bounded domain provided that the size e of the perturbation is sufficiently small this extends a result by d salamon in 25 according to which we do have the persistence of a single kam torus in the same framework moreover as for the persistence of a single torus the regularity assumption is essentially optimal

effective stability and kam theory abstract the two main stability results for nearly integrable hamiltonian systems are revisited nekhoroshev theorem concerning exponential lower bounds for the stability time effective stability and kam theorem concerning the preservation of a majority of the nonresonant invariant tori perpetual stability to stress the relationship between both theorems a common approach is given to their proof consisting of bringing the system to a normal form constructed through the lie series method the estimates obtained for the size of the remainder rely on bounds of the associated vectorfields allowing one to get the optimal stability exponent in nekhoroshev theorem for quasiconvex systems on the other hand a direct and complete proof of the isoenergetic kam theorem is obtained moreover a modification of the proof leads to the notion of nearly invariant torus which constitutes a bridge between kam and nekhoroshev theorems

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

results on normal forms for fpu chains in this paper we prove among other results that near the equilibirum position any periodic fpu chain with an odd number n of particles admits a birkhoff normal form up to order 4 whereas any periodic fpu chain with n even admits a resonant normal form up to order 4 this resonant normal form of order 4 turns out to be completely integrable further for n odd we obtain an explicit formula of the hessian of its hamiltonian at the fixed point

the spectrum of relativistic atoms according to bethe and salpeter and beyond we review evans contributions to the spectral theory of operators describing relativistic particle systems we will concentrate on no pair operators and recent extensions of that work

the quasi periodic centre saddle bifurcation abstract nearly integrable families of hamiltonian systems are considered in the neighbourhood of normally parabolic invariant tori in the integrable case such tori bifurcate into normally elliptic and normally hyperbolic invariant tori with a kam theoretic approach it is shown that both the normally parabolic tori and the bifurcation scenario survive a non integrable perturbation parametrised by pertinent large cantor sets these results are applied to rigid body dynamics