morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity

self dual instantons and holomorphic curves a gradient flow of a morse function on a compact riemannian manifold is said to be of morse smale type if the stable and unstable manifolds of any two critical points intersect transversally for such a morse smale gradient flow there is a chain complex generated by the critical points and graded by the morse index the boundary operator has as its x y entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the morse indices is 1 the homology of this chain complex agrees with the homology of the underlying manifold m and this can be used to prove the morse inequalities cf 33 26 around 1986 floer generalized this idea to infinite dimensional variational problems in which every critical point has infinite morse index but the moduli spaces of connecting orbits form finite dimensional manifolds for every pair of critical points the dimensions of these spaces give rise to a relative morse index and the boundary operator is defined by counting connecting

a superquadratic indefinite elliptic system and its morse conley floer homology we study critical points of the indefinite functional f h u v int nabla u cdot nabla v h x u v mathrm d x by applying floer s homology construction to the ordinary gradient flow of the functional f on a suitable sobolev space one of our main observations is that even though this flow is well posed in both time directions and lacks any kind of smoothing property one can still obtain compactness of connecting orbit spaces and thus define the floer homology for f h

on the morse index of lagrangian systems we give a new proof of the identity between the morse index of a periodic solution of a lagrangian system positive definite in the velocities and its maslov index furthermore we give an interpretation of the maslov index when the lagrangian system is not positive definite in the velocities our approach consists in relating the morse index of the solution to a relative morse index obtained passing to the hamiltonian formulation the main concepts used are the notion of commensurable subspaces relative dimension fredholm pairs and relative index of strongly indefinite bilinear forms

deformations of associative submanifolds with boundary abstract let m be a topological g 2 manifold we prove that the space of infinitesimal associative deformations of a compact associative submanifold y with boundary in a coassociative submanifold x is the solution space of an elliptic problem for a connected boundary y of genus g the index is given by y c 1 x 1 g where x denotes the orthogonal complement of t y in t x y and c 1 x the first chern class of x with respect to its natural complex structure further we exhibit explicit examples of non trivial index

subharmonic solutions of hamiltonian equations on tori let the torus t2n be equipped with the standard symplectic structure and a periodic hamiltonian dt e c3 sx x t2n r we look for periodic orbits of the hamiltonian flow u t j vdt t a subharmonic solution is a periodic orbit with minimal period an integral multiple m of the period of dt with m 1 we prove that if the hamiltonian flow has only finitely many orbits with the same period as dt then there are subharmonic solutions with arbitrarily high minimal period thus there are always infinitely many distinct periodic orbits the results proved here were proved in the nondegenerate case by conley and zehnder and in the case n 1 by le calvez

structure and stability of finite dimensional approximations for functional differential equations this paper deals with the structural and stability properties of the averaging approximation scheme for linear retarded functional differential equations both in the discrete and in the continuous time case the structure of the approximating systems is shown to be analogous to the structure of the underlying retarded equation moreover it is shown that the approximating systems are exponentially stable in a uniform sense if the original system is asymptotically stable

high action orbits for tonelli lagrangians and superlinear hamiltonians on compact configuration spaces abstract multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds given by lagrangians or hamiltonians which have quadratic growth in the velocities or in the momenta such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions in this note we show how these results can be extended to the classical setting of tonelli lagrangians lagrangians which are c 2 convex and superlinear in the velocities or to hamiltonians which are superlinear in the momenta and have a coercive action integrand

reeb orbits and the minimal discrepancy of an isolated singularity let a be an affine variety inside a complex n dimensional vector space which has an isolated singularity at the origin the intersection of a with a very small sphere turns out to be a contact manifold called the link of a any contact manifold contactomorphic to the link of a is said to be milnor fillable by a if the first chern class of our link is torsion then we can assign an invariant of our singularity called the minimal discrepancy which is an important invariant in birational geometry we define an invariant of the link up to contactomorphism using conley zehnder indices of reeb orbits and then we relate this invariant with the minimal discrepancy as a result we show that the standard contact five dimensional sphere has a unique milnor filling up to normalization proving a conjecture by seidel

an index theorem for non periodic solutions of hamiltonian systems we consider a hamiltonian setup m h l p where m is a symplectic manifold l is a distribution of lagrangian subspaces in m p a lagrangian submanifold of m h is a smooth time dependent hamiltonian function on m and a b m is an integral curve of the hamiltonian flow h starting at p we do not require any convexity property of the hamiltonian function h under the assump tion that b is not p focal it is introduced the maslov index imaslov of given in terms of the first relative homology group of the lagrangian g rassmannian under generic circumstances imaslov is computed as a sort of algebraic count of the p focal points along we prove the following version of the index theorem under suitable hypothe ses the morse index of the lagrangian action functional restricted to suitable variations of is equal to the sum of imaslov and a convexity term of the hamiltonian h rela tive to the submanifold p when the result is applied to the case of the cotangent bundle m t mof a semi riemannian manifold m g and to the geodesic hamiltonian h q p 1 g 1 p p we obtain a semi riemannian version of the celebrated morse index theorem for geodesics with variable endpoints in riemannian geometry

subharmonic solutions of hamiltonian systems and the maslov type index theory abstract this paper deals with the subharmonic solutions of hamiltonian systems h x j h t x t r 1 x r 2 n where h t x 1 2 b t x x h t x is t periodic in t b t is a t periodic semipositive symmetric matrix we prove that for each positive integer k h has a nonconstant kt periodic solution x k such that x j and x p j are geometrically distinct if p satisfies the certain conditions

on equivalence of floer s and quantum cohomology we show that the floer cohomology and quantum cohomology rings of the almost kahler manifoldm both defined over the novikov ring of the loop space m are isomorphic we do it using a brst trivial deformation of the topological a model the relevant aspect of noncompactness of the moduli of pseudoholomorphic instantons is discussed it is shown nonperturbatively that any brst trivial deformation of a model which does not change the dimensions of brst cohomology does not change the topological correlation functions either

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

symplectic rigidity symplectic fixed points and global perturbations of hamiltonian systems in this paper we study a generalized symplectic fixed point problem first considered by j moser in 20 from the point of view of some relatively recently discovered symplectic rigidity phenomena this problem has interesting applications concerning global perturbations of hamiltonian systems 2007 wiley periodicals inc

braid floer homology abstract area preserving diffeomorphisms of a 2 disc can be regarded as time 1 maps of non autonomous hamiltonian flows on r z d 2 the periodic flow lines define braid conjugacy classes up to full twists we examine the dynamics relative to such braid classes and define a new invariant for such classes the braid floer homology this refinement of floer homology originally used for the arnol d conjecture yields a morse type forcing theory for periodic points of area preserving diffeomorphisms of the 2 disc based on braiding contributions of this paper include 1 a monotonicity lemma for the behavior of the nonlinear cauchy riemann equations with respect to algebraic lengths of braids 2 establishment of the topological invariance of the resulting braid floer homology 3 a shift theorem describing the effect of twisting braids in terms of shifting the braid floer homology 4 computation of examples and 5 a forcing theorem for the dynamics of hamiltonian disc maps based on braid floer homology

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

spectral flow and bifurcation of critical points of strongly indefinite functionals abstract our main results here are as follows let x be a family of 2 periodic hamiltonian vectorfields that depend smoothly on a real parameter in a b and has a known trivial branch s of 2 periodic solutions let p be the poincare map of the linearization of x at s if the conley zehnder index of the path p does not vanish then any neighborhood of the trivial branch of periodic solutions contains 2 periodic solutions not on the branch moreover if each solution s is constant and each linearization a of x at s is time independent then bifurcation of 2 periodic orbits from the branch of equilibria arises whenever i a b i a b where i a is the index of the linear hamiltonian system ju au

periodic orbits of a one dimensional non autonomous hamiltonian system in this paper we study the properties of the periodic orbits of x vx t x 0 with x s1 and vx t x a t0 periodic potential called 1t0q the frequency of windings of an orbit in s1 we show that exists an infinite number of periodic solutions with a given we give a lower bound on the number of periodic orbits with a given period and by means of the morse theory

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

multiple periodic points of the poincare map of lagrangian systems on tori in this paper suppose l t x p 1 over 2 a x p cdot p v t x a is positive definite and symmetric and both a and v are c 3 and 1 periodic in all of their variables we prove that the poincare map i e the time 1 solution map of the lagrangian system

a relative morse index for the symplectic action the notion of a morse index of a function on a finite dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold in this paper we define for any pair of critical points of a a relative morse index which corresponds to the difference of the two morse indices in finite dimensions it is based on the spectral flow of the hessian of a and can be identified with a topological invariant recently defined by viterbo and with the dimension of the space of trajectories between the two critical points

a module structure on the symplectic floer cohomology the aim of this paper is to offer an affirmative answer to the floer conjectures in 2 p 589 which states that there is a module structure on the z 2 n graded symplectic floer cohomology for monotone symplectic manifolds by constructing a z graded symplectic floer cohomology as an integral lift of the z 2 n graded symplectic floer cohomology we gain control of the holomorphic bubbling spheres this makes a module structure on the z graded floer cohomology there is a spectral sequence with e 1 given by the z graded symplectic floer cohomology such a spectral sequence converges to the z 2 n graded symplectic floer cohomology hence we induce a module structure for the z 2 n graded symplectic floer cohomology by the spectral sequence and algebraic topology methods

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

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

growth of maps distortion in groups and symplectic geometry in the present paper we study two sequences of real numbers associated to a symplectic diffeomorphism the uniform norm of the differential of its n th iteration the word length of its n th iteration where we assume that our diffeomorphism lies in a finitely generated group of symplectic diffeomorphisms we find lower bounds for the growth rates of these sequences in a number of situations these bounds depend on the symplectic geometry of the manifold rather than on the specific choice of a diffeomorphism they are obtained by using recent results of schwarz on floer homology as an application we prove non existence of certain non linear symplectic representations for finitely generated groups

on the arnold conjecture for weakly monotone symplectic manifolds we show the arnold conjecture concerning symplectic fixed points in the case that the symplectic manifold is weakly monotone and all the fixed points are non degenerate in particular the conjecture is true in dimension 2 4 6 if all the fixed points are non degenerate

spectral flow crossing forms and homoclinics of hamiltonian systems we prove a spectral flow formula for one parameter families of hamiltonian systems under homoclinic boundary conditions which relates the spectral flow to the relative maslov index of a pair of curves of lagrangians induced by the stable and unstable subspaces respectively finally we deduce sufficient conditions for bifurcation of homoclinic trajectories of one parameter families of nonautonomous amiltonian vector fields

witten s complex and infinite dimensional morse theory we investigate the relation between the trajectories of a finite dimensional gradient flow connecting two critical points and the cohomology of the surrounding space the results are applied to an infinite dimensional problem involving the symplect c action function

multiple brake orbits in bounded convex symmetric domains abstract in this paper we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in r n for n 2

gromov s compactness of pseudo holomorphic curves and symplectic geometry on utilise certaines idees de schoerr et uhlenbeck sur les equations aux derivees partielles pour demontrer la compacite de gromov des courbes pseudoholomorphes

k area hofer metric and geometry of conjugacy classes in lie groups given a closed symplectic manifold m we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group ham m by means of the hofer metric on ham m we use pseudo holomorphic curves involved in the definition of the multiplicative structure on the floer cohomology of a symplectic manifold m to estimate this quantity in terms of actions of some periodic orbits of related hamiltonian flows as a corollary we get a new way to obtain agnihotri belkale woodward inequalities for eigenvalues of products of unitary matrices as another corollary we get a new proof of the geodesic property with respect to the hofer metric of hamiltonian flows generated by certain autonomous hamiltonians our main technical tool is k area defined for hamiltonian fibrations over a surface with boundary in the spirit of l polterovich s work on hamiltonian fibrations over s2

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations we study pseudoholomorphic maps from a punctured riemann surface into the symplectization of a contact manifold a fredholm theory yields the virtual dimension of the moduli spaces of such maps in terms of the euler characteristic of the riemann surface and the asymptotics data given by the periodic solutions of the reeb vector field associated to the contact form the transversality results establish the existence of additional structure for these spaces to be more precise we prove that these spaces are generically smooth manifolds and therefore their virtual dimension coincides with their actual dimension

inexact restoration method for derivative free optimization with smooth constraints a new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not the method is based on the inexact restoration framework by means of which each iteration is divided in two phases in the first phase one considers only the constraints in order to improve feasibility in the second phase one minimizes a suitable objective function subject to a linear approximation of the constraints the second phase must be solved using derivative free methods an algorithm introduced recently by kolda lewis and torczon for linearly constrained derivative free optimization is employed for this purpose under usual assumptions convergence to stationary points is proved a computer implementation is described and numerical experiments are presented

finite energy pseudoholomorphic planes with multiple asymptotic limits it s known from ann inst h poincare anal non lineaire 13 3 337 379 11 properties of pseudoholomorphic curves in symplectisation iv asymptotics with degeneracies in contact and symplectic geometry cambridge 1994 vol 8 publ newton inst pp 78 117 cambridge univ press cambridge 10 a morse bott approach to contact homology phd thesis stanford university 2 that in a contact manifold equipped with either a nondegenerate or morse bott contact form a finite energy pseudoholomorphic curve will be asymptotic at each of its nonremovable punctures to a single periodic orbit of the reeb vector field and that the convergence is exponential we provide examples here to show that this need not be the case if the contact form is degenerate more specifically we show that on any contact manifold m xi with cooriented contact structure one can choose a contact form lambda with ker lambda xi and a compatible complex structure j on xi so that for the associated mathbb r invariant almost complex structure tilde j on mathbb r times m there exist families of embedded finite energy tilde j holomorphic cylinders and planes having embedded tori as limit sets

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

the conley conjecture for hamiltonian systems on the cotangent bundle and its analogue for lagrangian systems abstract in this paper the conley conjecture which was recently proved by franks and handel j franks m handel periodic points of hamiltonian surface diffeomorphism geom topol 7 2003 713 756 for surfaces of positive genus hingston n hingston subharmonic solutions of hamiltonian equations on tori ann math in press for tori and ginzburg v l ginzburg the conley conjecture arxiv math sg 0610956v1 for closed symplectically aspherical manifolds is proved for c 1 hamiltonian systems on the cotangent bundle of a c 3 smooth compact manifold m without boundary of a time 1 periodic c 2 smooth hamiltonian h r t m r which is strongly convex and has quadratic growth on the fibers namely we show that such a hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations if h also satisfies h t q p h t q p for any t q p r t m it is shown that the time 1 map of the hamiltonian system if exists has infinitely many periodic points siting in the zero section of t m if m is c 5 smooth and dim m 1 h is of c 4 class and independent of time t then for any 0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of such that any one of them cannot be obtained from others by iterations or rotations these results are obtained by proving similar results for the lagrangian system of the fenchel transform of h l r t m r which is proved to be strongly convex and to have quadratic growth in the velocities yet

the poincare hopf theorem for relative braid classes this paper studies the existence and multiplicity of closed integral curves of smooth vector fields on the closed 2 disc mathbb d 2 in the complement of a suitable geometric braid y called a skeleton the strands in a skeleton y must be trajectories of the vector field and the 1 periodic orbits to be counted must also be representatives of a suitable relative braid class x in the complement of y from leray schauder degree theory a signed count of the closed trajectories representing x in the complement of y is independent of the vector field it therefore suffices to compute or estimate the signed count of periodic orbits for a particularly convenient vector field we show that the signed count equals the euler floer characteristic of braid floer homology cf van den berg et al in j differ equ 259 5 1663 1721 2015 the latter can be computed via a finite cube complex which serves as a model for the given braid class

propagation in hamiltonian dynamics and relative symplectic homology the main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating hamiltonian is sufficiently large over the zero section the proof is based on floer homology and on the notion of a relative symplectic capacity applications include results about propagation properties of sequential hamiltonian systems periodic orbits on hypersurfaces hamiltonian circle actions and smooth lagrangian skeletons in stein manifolds

index iteration theories for periodic orbits old and new abstract in this paper we give a survey of the index iteration theories developed for periodic solution orbits of hamiltonian systems in treating various problems from 1950s up to now including some recent developments and some of their applications to periodic solution problems

lusternik schnirelmann theory and closed reeb orbits we develop a variant of lusternik schnirelmann theory for the shift operator in equivariant floer and symplectic homology our key result is that the spectral invariants are strictly decreasing under the action of the shift operator when periodic orbits are isolated as an application we prove new multiplicity results for simple closed reeb orbits on the standard contact sphere the unit cotangent bundle to the sphere and some other contact manifolds we also show that the lower conley zehnder index enjoys a certain recurrence property and revisit and reprove from a different perspective a variant of the common jump theorem of long and zhu this is the second combinatorial ingredient in the proof of the multiplicity results

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

transversality for local morse homology with symmetries and applications we prove the transversality result necessary for defining local morse chain complexes with finite cyclic group symmetry our arguments use special regularized distance functions constructed using classical covering lemmas and an inductive perturbation process indexed by the strata of the isotropy set a global existence theorem for symmetric morse smale pairs is also proved regarding applications we focus on hamiltonian dynamics and rigorously establish a local contact homology package based on discrete action functionals we prove a persistence theorem analogous to the classical shifting lemma for geodesics asserting that the iteration map is an isomorphism for good and admissible iterations we also consider a chas sullivan product on non invariant local morse homology which plays the role of pair of pants product and study its relationship to symplectically degenerate maxima finally we explore how our invariants can be used to study bifurcation of critical points and periodic points under additional symmetries

on the floer homology of cotangent bundles this paper concerns floer homology for periodic orbits and for a lagrangian intersection problem on the cotangent bundle t m of a compact orientable manifold m the first result is a new l estimate for the solutions of the floer equation which allows us to deal with a larger and more natural class of hamiltonians the second and main result is a new construction of the isomorphism between the floer homology and the singular homology of the free loop space of m in the periodic case or of the based loop space of m in the lagrangian intersection problem the idea for the construction of such an isomorphism is to consider a hamiltonian that is the legendre transform of a lagrangian on t m and to construct an isomorphism between the floer complex and the morse complex of the classical lagrangian action functional on the space of w1 2 free or based loops on m 2005 wiley periodicals inc

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

dynamical convexity and elliptic periodic orbits for reeb flows a long standing conjecture in hamiltonian dynamics states that the reeb flow of any convex hypersurface in mathbb r 2n carries an elliptic closed orbit two important contributions toward its proof were given by ekeland an index theory for periodic solutions of convex hamiltonian systems 1986 and dell antonio d onofrio ekeland periodic solutions of elliptic type for strongly nonlinear hamiltonian systems 1995 proving this for convex hypersurfaces satisfying suitable pinching conditions and for antipodal invariant convex hypersurfaces respectively in this work we present a generalization of these results using contact homology and a notion of dynamical convexity first introduced by hofer wysocki zehnder for tight contact forms on s 3 applications include geodesic flows under pinching conditions magnetic flows and toric contact manifolds

symplectically degenerate maxima via generating functions we provide a simple proof of a theorem due to nancy hingston asserting that symplectically degenerate maxima of any hamiltonian diffeomorphism phi of the standard symplectic 2d torus are non isolated contractible periodic points or their action is a non isolated point of the average action spectrum of phi our argument is based on generating functions

floer theory for negative line bundles via gromov witten invariants abstract we prove that the gw theory of negative line bundles m tot l b determines the symplectic cohomology indeed s h m is the quotient of q h m by the kernel of a power of quantum cup product by c 1 l we prove this also for negative vector bundles and the top chern class we calculate s h and q h for o n c p m for example for o 1 m is the blow up of c m 1 at the origin and s h m has rank m we prove kodaira vanishing for very negative l s h 0 and serre vanishing if we twist a complex vector bundle by a large power of l s h 0 observe s h m 0 iff c 1 l is nilpotent in q h m this implies oancea s result b 2 b 0 s h m 0 we prove the weinstein conjecture for any contact hypersurface surrounding the zero section of a negative line bundle for symplectic manifolds x conical at infinity we build a homomorphism from 1 ham l x to invertibles in s h x this is similar to seidel s representation for closed x except now they are not invertibles in q h x