the denjoy wolff theorem for condensing holomorphic mappings if b is the open unit ball of a strictly convex banach space x and f b b is holomorphic condensing with respect to and fixed point free then there exists b such that the sequence lcub fn of the iterates of f converges in the compact open topology to the constant map taking the value

local existence of the borel transform in euclidean φ 4 4 we bound rigorously the large order behaviour of 44 euclidean perturbative quantum field theory as the simplest example of renormalizable but non super renormalizable theory the needed methods are developed to take into account the structure of renormalization which plays a crucial role in the estimates as a main thorem it is shown that the schwinger functions at ordern are bounded byknn which implies a finite radius of convergence for the borel transform of the perturbation series

backlund transformations and loop group actions we construct a local action of the group of rational maps from 2 to gl n on local solutions of flows of the zs akns sl n hierarchy we show that the actions of simple elements linear fractional transformations give local backlund transformations and we derive a permutability formula from different factorizations of a quadratic element we prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities however the action of the subgroup of the rational maps that satisfy the u n reality condition on the space of global rapidly decaying solutions of the flows in the u n hierarchy is global and the action of a simple element gives a global backlund transformation the actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multisolitons multibreathers we show that this theory generalizes the classical backlund theory of the sine gordon equation the group structures of backlund transformations for various hierarchies are determined by their reality conditions we identify the reality conditions the group structures for the sl n u k n k kdv kupershmidt wilson and gel fand dikii hierarchies the actions of linear fractional transformations that satisfy a reality condition modulo the center of the group of rational maps give backlund and darboux transformations for the hierarchy defined by the reality condition since the factorization cannot always be carried out under this reality condition the action is again local and backlund transformations only generate local solutions for these hierarchies unless singular solutions are allowed 2000 john wiley sons inc

on the large order behavior of φ44 we continue the rigorous study of the large order behavior of the perturbation series for the 4 model in 4 dimensions started in 1 in this paper we prove a result announced in 1 we show that the exact radius of convergence of the borel transform of the renormalized perturbation series for 44 is greater than or equal to the expected value given by the position of the first renormalon 2 this result holds for any vector 2 2 model withn components and makes use of the lipatov bound of 1 this result is based on a partial resummation of counterterms similar to the one of 3 but in a phase space analysis of the renormalized series

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

inequalities for proper contractions and strictly dissipative operators abstract some inequalities for proper contractions and strictly dissipative operators on a hilbert space are proved these inequalities for operators are related to the classical harnack s inequalities for analytic functions with a positive real part

operator valued typically real functions induced by a contraction let h be a complex hilbert space and let l h denote the algebra of all bounded linear operators on h letf z n 0 znanbe an operator valued analytic function whose coefficients are bounded operators in l h forzin the open unit disc d and the series is convergent in the strong operator topology in this paper we discuss operator valued typically real functionsf z n 0 znanwhich generalize complex valued typically real functions we characterize operator valued typically real functions and study such functionsf z induced by a contraction on h in addition we considerm n tuple operator valued typically real functions and positively real functions finally we charaterize operator valued typically real functions in the finite dimensional cases

large order estimates for ground state energy perturbation series a general treatment is given using path integral methods of obtaining accurate estimates on the rate of growth at large order of the perturbation coefficients for the lowest eigenvalue ground state energy of a large class of anharmonic oscillators simple sufficient conditions are given on the potentialv x so that accurate upper and lower bounds on the perturbation coefficients may be derived several examples are given which generalize previous results examples from euclidean quantum field theory are also considered

on the local borel transform of perturbation theory we prove existence of the local borel transform for the perturbative series of massive varphi 4 4 theory as compared to previous proofs in the literature the present bounds are much sharper as regards the dependence on external momenta they are explicit in the number of external legs and they are obtained quite simply through a judiciously chosen induction hypothesis applied to the wegner wilson polchinski flow equations we pay attention not to generate an astronomically large numerical constant for the inverse radius of convergence of the borel transform

the lipatov argument for φ 3 4 perturbation theory we extend to 3 4 the work of s breen on the leading behavior at large order of 2 4 perturbation theory using a phase space expansion to obtain new estimates on the high energy behavior of 3 4 feynman graphs and a rigorous semiclassical expansion we prove that the radius of convergence of the borel transform of the pertubative series for 3 4 euclidean field theory is the one computed by the lipatov method

isosystolic inequalities and the topological expansion for random surface and matrix models using the isosystolic inequalities on riemann surfaces we prove that for many random surface or matrix models the radius of convergence of the perturbative series at fixed genus is independent of the genus this result applies for instance to the dynamically triangulated random surface model in any dimension or to many matrix models with regular propagators in the superrenormalizable domain for instance 3 in dimensiond 6 lambda phi 4 sqrt lambda phi 3 in dimensiond 4 and various otherp 2 models in particular all those containing an odd power of we hope that this result is a first step towards a more rigorous understanding of the genus dependence of surface models or of quantum gravity coupled with matter fields

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

iteration of holomorphic maps in hilbert spaces abstract previous work gong ning chen j math anal appl 98 1984 305 313 on iteration of holomorphic maps of cn is continued the purpose of this note is to extend results given in the above mentioned reference to the case of complex hilbert spaces other comments are appended

construction and borel summability of planar 4 dimensional euclidean field theory we use the methods of 1 to show that the planar part of the renormalized perturbation theory for 4 4 euclidean field theory is borel summable on the asymptotically free side of the theory the borel sum can therefore be taken as a rigorous definition of then limit of a massiven n matrix model with a trg 4 interaction hence with wrong sign ofg our construction is relevant for a solution of the ultra violet problem for planar qcd we also propose a program for studying the structure of the renormalons singularities within the planar world

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

a lipatov bound for φ 4 4 euclidean field theory we bound the large order behavior of these pieces of the renormalized perturbation expansion for 4 4 which do not contain renormalons 1 the bound we obtain has the form of the leading asymptotic behavior computed by the lipatov method with the exact value of the lipatov constant therefore this paper is a step towards the rigorous study of the large order behavior of 4 4 and towards a proof of existence of the first renormalon singularity which should prevent the theory from being borel summable using the results of this paper and the techniques of 15 one can for instance improve 17 the estimate 18 on the finiteness of the radius of convergence of the borel transform of renormalized 4 4 and obtain that this radius is at least the exact value conjectured in 1

leading large order asymptotics for φ4 2 perturbation theory we develop a rigorous semiclassical expansion to compute the radius of convergence of the borel transform for the pressure in 4 2 field theory this result gives a partial justification for the lipatov method of finding large order perturbation theory asymptotics in quantum field theory

breathers as homoclinic geometric wave maps a classical breather solution of the sine gordon equation can be viewed as a homoclinic wave map from periodic minkowski space mp2 into s2 this wave map is an unstable orbit of a simple steady state wave map homoclinic wave maps are also constructed into non round spheres

bounds on completely convergent euclidean feynman graphs letg be a euclidean feynman graph containingl g lines we prove that ifg has massive propagators and does not contain any divergent subgraphs its value is bounded bykl g we also prove the infrared analogue of this bound

random iterations fixed points and invariant crf horospheres in complex banach spaces abstract for holomorphic noncontractive maps on not necessarily bounded domains in complex banach spaces we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations in particular we show that the problem concerning the existence of the horospheres determined by caratheodory reiffen finsler pseudometrics defined on unbounded domains has the solution and we prove new results of type of julia s lemma and wolff s theorem

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

resonances radiation damping and instabilitym in hamiltonian nonlinear wave equations we consider a class of nonlinear klein gordon equations which are hamiltonian and are perturbations of linear dispersive equations the unperturbed dynamical system has a bound state a spatially localized and time periodic solution we show that for ge neric nonlinear hamiltonian perturbations all small amplitude so lutions decay to zero as time tends to infinity at an anomalously slow rate in particular spatially localized and time periodic solutions of the linear problem are destroyed by generic nonlinear hamiltonian perturbations via slow radiation of energy to infinity these solutions can therefore be thought of as metastable states the main mecha nism is a nonlinear resonant interaction of bound states eigenfunc tions and radiation continuous spectral modes leading to energy transfer from the discrete to continuum modes this is in contrast to the kam theory in which appropriate nonresonance conditions im ply the persistence of invariant tori a hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the fermi golden rule arising in the theory of resonances in quantum mechanics the techniques used involve i a time dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues ii a generalization of the hamiltonian normal form appropriate for infinite dimensional dis persive systems and iii ideas from scattering theory the arguments are quite general and we expect them to apply to a large class of

phase space analysis and particle structure in field theory progress in the last few years in constructive field theory has included 1 the construction of three dimensional non abelian gauge theories in a finite volume 2 the construction of models that are asymptotically free or have a nontrivial fixed point 3 a new presentation of perturbation theory yielding old and new large order estimates and 4 asymptotic completeness for renormalizable models in the two particle region the existence of multiparticle structure equations in the euclidean region provided that the coupling constant goes to zero as the energy increases these equations yield formally the asymptotic completeness in obtaining most of these results the phase space expansion pioneered by glimm and jaffe plays a crucial role the first part of this review is a description of the phase space expansion in the ecole polytechnique version the second part is devoted to the construction of models the third part treats perturbation theory and the last part deals with asymptotic completeness

weak solutions and development of singularities of the su 2 σ model pour toutes donnees initiales d energie finie le modele sigma o k a des solutions faibles globales en chaque dimension d espace temps

random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces conditions which guarantee the uniform convergence of random iterations of holomorphic contractions on unbounded domains in complex hausdorff locally convex and sequentially complete topological vector spaces are established also conditions concerning the convergence of random iterations of weaker contractions in hausdorff uniform spaces are given

a class of overdetermined systems defined by tableaux involutiveness and the cauchy problem this article addresses the question of involutiveness and discusses the initial value problem for a class of overdetermined systems of partial differential equations which arise in the theory of integrable systems and are defined by tableaux