index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

iteration inequalities of the maslov type index theory with applications abstract in this paper by using the index theory introduced by y long in 1999 pacific j math 187 113 149 in particular the splitting numbers maslov type mean index and the homotopy component of symplectic matrix we establish various inequalities of the maslov type index theory for iterations of symplectic paths starting from the identity as an application these results are used to study rabinowitz conjecture on the prescribed minimal period solution problem of nonlinear hamiltonian systems

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

branching processes in levy processes laplace functionals of snakes and superprocesses we use the exploration process introduced in a previous work to develop a new construction of superprocesses with a general branching mechanism this construction depends on a path valued process called the levy snake which is of independent interest our method of proof involves a calculation of the laplace functional of the occupation field of the levy snake this calculation relies on an evaluation of the corresponding moment functionals which requires precise information about the underlying genealogical structure

linear stability of the elliptic lagrangian triangle solutions in the three body problem abstract this paper concerns the linear stability of the well known periodic orbits of lagrange in the three body problem given any three masses there exists a family of periodic solutions for which each body is at the vertex of an equilateral triangle and travels along an elliptic kepler orbit reductions are performed to derive equations which determine the linear stability of the periodic solutions these equations depend on two parameters the eccentricity e of the orbit and the mass parameter 27 m1m2 m1m3 m2m3 m1 m2 m3 2 a combination of numerical and analytic methods is used to find the regions of stability in the e plane in particular using perturbation techniques it is rigorously proven that there are mass values where the truly elliptic orbits are linearly stable even though the circular orbits are not

existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem abstract in this paper we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four body problem the hamiltonian of the differential system is regularized by a levi civita type transformation and an appropriate scaling of time the initial condition of this orbit is shown to be the infimum of some well chosen set this existence proof is direct and surprisingly simple further a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group d 4 then roberts symmetry reduction method is applied to show the linear stability it turns out that the rhomboidal periodic orbit in the planar equal mass four body problem is linearly stable

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

closed characteristics on compact convex hypersurfaces in r 2n for any given compact c 2 hypersurface in r 2n bounding a strictly convex set with nonempty interior in this paper an invariant n is defined and satisfies n n 2 1 where a denotes the greatest integer which is not greater than a r the following results are proved in this paper there always exist at least n geometrically distinct closed characteristics on if all the geometrically distinct closed characteristics on are nondegenerate then n n if the total number of geometrically distinct closed characteristics on is finite there exists at least an elliptic one among them and there exist at least n 1 of them possessing irrational mean indices if this total number is at most 2 n 2 there exist at least two elliptic ones among them

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

branching processes in levy processes the exploration process the main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure the exploration process appears as a simple local time functional of a levy process with no negative jumps whose laplace exponent coincides with the branching mechanism function this new relation between spectrally positive levy processes and continuous branching processes provides a unified perspective on both theories in particular we derive the adequate formulation of the classical ray knight theorem for such levy processes as a consequence of this theorem we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process

the uniform random tree in a brownian excursion to any brownian excursione with duration e and anyt 1 t p 0 e we associate a branching tree withp branches denoted byt p e t 1 t p which is closely related to the structure of the minima ofe our main theorem states that ife is chosen according to the ito measure and t 1 t p according to lebesgue measure on 0 e p the treet p e t 1 t p is distributed according to the uniform measure on the set of trees withp branches the proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree thus generalizing a well known representation theorem of bismut if we replace the ito measure by the law of the normalized excursion a simple conditioning argument leads to another remarkable result originally proved by aldous with a very different method

an investigation into two bin packing problems with ordering and orientation implications this paper considers variants of the one dimensional bin packing and stock cutting problem in which both the ordering and orientation of items in a container influences the validity and quality of a solution two new real world problems of this type are introduced the first that involves the creation of wooden trapezoidal shaped trusses for use in the roofing industry the second that requires the cutting and scoring of rectangular pieces of cardboard in the construction of boxes to tackle these problems two variants of a local search based approximation algorithm are proposed the first that attempts to determine item ordering and orientation via simple heuristics the second that employs more accurate but costly branch and bound procedures we investigate the inevitable trade off between speed and accuracy that occurs with these variants and highlight the circumstances under which each scheme is advantageous

self similar fragmentations derived from the stable tree i the basic object we consider is a certain model of continuum random tree called the stable tree we construct a fragmentation process f t t 0 out of this tree by removing the vertices located under height t thanks to a self similarity property of the stable tree we show that the fragmentation process is also self similar the semigroup and other features of the fragmentation are given explicitly asymptotic results are given as well as a couple of related results on continuous state branching processes

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

probabilistic and fractal aspects of levy trees we investigate the random continuous trees called levy trees which are obtained as scaling limits of discrete galton watson trees we give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted trees which is equipped with the gromov hausdorff distance to construct levy trees we make use of the coding by the height process which was studied in detail in previous work we then investigate various probabilistic properties of levy trees in particular we establish a branching property analogous to the well known property for galton watson trees conditionally given the tree below level a the subtrees originating from that level are distributed as the atoms of a poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root we study regularity properties of local times in the space variable and prove that the support of local time is the full level set except for certain exceptional values of a corresponding to local extinctions we also compute several fractal dimensions of levy trees including hausdorff and packing dimensions in terms of lower and upper indices for the branching mechanism function which characterizes the distribution of the tree we finally discuss some applications to super brownian motion with a general branching mechanism

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

self similar fragmentations derived from the stable tree ii splitting at nodes we study a natural fragmentation process of the so called stable tree introduced by duquesne and le gall which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self similar with positive index explicit formulas for the semigroup are given and we provide asymptotic results we also give an alternative construction of this fragmentation using paths of levy processes hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the brownian continuum random tree or using brownian paths respectively due to aldous pitman and bertoin

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

scaling limits of random planar maps with large faces we discuss asymptotics for large random planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with index alpha in 1 2 when the number n of vertices of the map tends to infinity the asymptotic behavior of distances from a distinguished vertex is described by a random process called the continuous distance process which can be constructed from a centered stable process with no negative jumps and index alpha in particular the profile of distances in the map rescaled by the factor n 1 2 alpha converges to a random measure defined in terms of the distance process with the same rescaling of distances the vertex set viewed as a metric space converges in distribution as n to infty at least along suitable subsequences towards a limiting random compact metric space whose hausdorff dimension is equal to 2 alpha

sub gaussian tail bounds for the width and height of conditioned galton watson trees we study the height and width of a galton watson tree with offspring distribution satisfying e 1 0 var conditioned on having exactly n nodes under this conditioning we derive sub gaussian tail bounds for both the width largest number of nodes in any level and height greatest level containing a node the bounds are optimal up to constant factors in the exponent under the same conditioning we also derive essentially optimal upper tail bounds for the number of nodes at level k for 1 k n

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

variational property of periodic kepler orbits in constant curvature spaces abstract we study the variational property of the periodic kepler orbits on the sphere the plane and the hyperbolic plane we first classify the orbits by the two constants of motion the energy and the angular momentum then we characterize the local variational property of the closed orbits by computing the maslov type indices finally we study the global variational property of the closed orbits we prove that the closed orbits on the hyperbolic plane minimizes the action among all loops which encircle the attracting center

a limit theorem for the contour process of condidtioned galton watson trees in this work we study asymptotics of the genealogy of galton watson processes conditioned on the total progeny we consider a fixed aperiodic and critical offspring distribution such that the rescaled galton watson processes converges to a continuous state branching process csbp with a stable branching mechanism of index 1 2 we code the genealogy by two different processes the contour process and the height process that le gall and le jan recently introduced we show that the rescaled height process of the corresponding galton watson family tree with one ancestor and conditioned on the total progeny converges in a functional sense to a new process the normalized excursion of the continuous height process associated with the stable csbp we deduce from this convergence an analogous limit theorem for the contour process in the brownian case 2 the limiting process is the normalized brownian excursion that codes the continuum random tree the result is due to aldous who used a different method

the real tree in this note it is shown that there exists a natural metric on the set formula of bounded subsets of r containing their infimum which endowstwith the structure of an r tree so that for everyt t the number of connected components oft t coincides with the cardinality p r of the set of subsets of r in addition the set of ends oftis explicitly determined and various further features oftare discussed too

random stable laminations of the disk we study large random dissections of polygons we consider random dissections of a regular polygon with n sides which are chosen according to boltzmann weights in the domain of attraction of a stable law of index 2 1 2 as n goes to infinity we prove that these random dissections converge in distribution towards a random compact set called the random stable lamination if 2 we recover aldous brownian triangulation however if 2 1 2 large faces remain in the limit and a dierent random compact set appears we show that the random stable lamination can be coded by the continuous time height function associated to the normalized excursion of a strictly stable spectrally positive levy process of index using this coding we establish that the hausdor dimension of the stable random lamination is almost surely 2 1

percolation on random triangulations and stable looptrees we study site percolation on angel and schramm s uniform infinite planar triangulation we compute several critical and near critical exponents and describe the scaling limit of the boundary of large percolation clusters in all regimes subcritical critical and supercritical we prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3 2 which was introduced in curien and kortchemski random stable looptrees arxiv 1304 1044 2014 we also give a conjecture linking looptrees of any index alpha in 1 2 with scaling limits of cluster boundaries in random triangulations decorated with o n models

scaling limits of random polya trees polya trees are rooted trees considered up to symmetry we establish the convergence of large uniform random polya trees with arbitrary degree restrictions to aldous continuum random tree with respect to the gromov hausdorff metric our proof is short and elementary and it is based on a novel decomposition it shows that the global shape of a random polya tree is essentially dictated by a large galton watson tree that it contains we also derive sub gaussian tail bounds for both the height and the width which are optimal up to constant factors in the exponent

limits of the boundary of random planar maps we discuss asymptotics for the boundary of critical boltzmann planar maps under the assumption that the distribution of the degree of a typical face is in the domain of attraction of a stable distribution with parameter alpha in 1 2 first in the dense phase corresponding to alpha in 1 3 2 we prove that the scaling limit of the boundary is the random stable looptree with parameter 1 alpha 1 2 second we show the existence of a phase transition through local limits of the boundary in the dense phase the boundary is tree like while in the dilute phase corresponding to alpha in 3 2 2 it has a component homeomorphic to the half plane as an application we identify the limits of loops conditioned to be large in the rigid o n loop model on quadrangulations proving thereby a conjecture of curien and kortchemski

continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models given any regularly varying dislocation measure we identify a natural self similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths as an application we obtain continuum random tree limits of aldous s beta splitting models and ford s alpha models for phylogenetic trees this confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf

elliptic relative equilibria in the n body problem a planar central configuration of the n body problem gives rise to a solution where each particle moves on a specific keplerian orbit while the totality of the particles move on a homothety motion if the keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium the totality of such solutions forms a four dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement in our coordinate system the linear variational equations of such a solution decouple into three subsystems one subsystem simply gives the motion of the center of mass another is kepler s problem and the third determines the nontrivial characteristic multipliers using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three body problem we reproduce the analytic studies of g roberts we also study the linear stability of the four and five body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or

sub exponential tail bounds for conditioned stable bienayme galton watson trees we establish uniform sub exponential tail bounds for the width height and maximal outdegree of critical bienayme galton watson trees conditioned on having a large fixed size whose offspring distribution belongs to the domain of attraction of a stable law this extends results obtained for the height and width by addario berry devroye and janson in the finite variance case

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

the stable trees are nested we show that we can construct simultaneously all the stable trees as a nested family more precisely if 1 alpha alpha prime le 2 we prove that hidden inside any alpha stable tree we can find a version of an alpha prime stable tree rescaled by an independent mittag leffler type distribution this tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it our proofs are based on a recursive construction due to marchal which is proved to converge almost surely towards a stable tree

geometric characterization for variational minimization solutions of the 3 body problem with fixed energy abstract in this paper for any given positive masses we prove that the constrained variational minimization solutions of the fixed energy or fixed mean potential energy problem for the 3 body type problems with homogeneous potentials in r 3 or r 2 are precisely the planar equilateral triangle circular solutions found by j lagrange in 1772

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion