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

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

associativity of quantum multiplication we proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold v for which c 1 a 1 for any effective class the same proof also works for any positive compact symplectic manifold with c 1 a 1

poincare birkhoff theorems in random dynamics we propose a generalization of the poincare birkhoff theorem on area preserving twist maps to area preserving twist maps f that are random with respect to an ergodic probability measure in this direction we will prove several theorems concerning existence density and type of the fixed points to this end first we introduce a randomized version of generalized generating functions and verify the correspondence between its critical points and the fixed points of f a fact which we successively exploit in order to prove the theorems the study we carry out needs to combine probabilistic techniques with methods from nonlinear pde and from differential geometry notably moser s method and conley zehnder theory our stochastic model in the periodic case coincides with the classical setting of the poincare birkhoff theorem

hierarchies of tree series transformations we study bottom up and top down tree series transducers over a semiring a and denote the tree series transformation classes computed by them by bott ts a and topt ts a respectively we present the inclusion diagram of the classes p bott tsn a p topt tsn a p botn 1t ts a and p topn 1t ts a and prove its correctness where a is a commutative izz semiring izz idempotent zero divisor free and zero sum free and the prefix p stands for polynomial this inclusion diagram implies the properness of the following four hierarchies p topt ts a p top2t ts a p top3t ts a p bott ts a p bot2t ts a p bot3t ts a p topt ts a p bot2t ts a p top3t ts a p bot4t ts a p bott ts a p top2t ts a p bot3t ts a p top4t ts a where the first hierarchy generalizes the famous top down tree transformation hierarchy of engelfriet math systems theory 15 1982 92 125 as the second main result we prove that the first two hierarchies are proper even for arbitrary i e not necessarily commutative izz semirings

existence characterizations of temporal safety supervisors this note shows that beyond the expected basic results on supervisor existence and synthesis the temporal logic systems analysis for the invariance control of an arbitrary past formula p yields new insights into supremal control of temporal safety these insights come in the form of equivalent temporal characterizations one characterization allows a natural language interpretation which provides a very good intuitive feel of the supremal controllability concept another provides a structurally elegant representation revealing that the past formula p must never be stronger than some fixed logic constant importantly from the control synthesis viewpoint these temporal characterizations provide the opportunity to use several industrial strength commercially available theorem provers to accomplish the synthesis task in a transparent fashion finally a controllable canonical form unifying the supremal exact and infimal existences of temporal safety control is exhibited and its theoretical significance is discussed

a method for the synthesis of controllers to handle safety liveness and real time constraints describes a synthesis method that automatically derives controllers for timed discrete event systems with nonterminating behavior modeled by timed transition graphs and specifications of control requirements expressed by metric temporal logic mtl formulas synthesis is performed by using 1 a forward chaining search that evaluates the satisfiability of mtl formulas over sequences of states generated by occurrences of actions and 2 a control directed backtracking technique that takes into consideration the controllability of actions this method has several interesting features first the issues of controllability safety liveness and real time are integrated in a single framework second the synthesis process does not require explicit storage of an entire transition structure over which formulas are checked and can be stopped at any moment giving an approximate but useful result third search and control mechanisms allow circumvention of the state explosion problem

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

syntax based synthesis for temporal safety supervision this paper presents a supremal control generalization of a temporal version of supervisory control for temporal safety in discrete event systems the generalization is an outcome of the development of a temporal logic equations approach formulated using unary transformer concepts borrowed mainly from the predicate and predicate transform theory of dijkstra and scholten the proposed syntax based approach has not hitherto been formally augmented to the temporal logic framework and is shown to parallel and to generalize the boolean equations approach proposed in the existing predicate control theory of kumar garg and marcus a detailed example which cannot be handled by the existing predicate control theory demonstrates how the natural language basis of temporal logic for requirement specification and syntax based control synthesis can be unified in a common logic framework

an exact sequence for contact and symplectic homology a symplectic manifold w with contact type boundary m w induces a linearization of the contact homology of m with corresponding linearized contact homology hc m we establish a gysin type exact sequence in which the symplectic homology sh w of w maps to hc m which in turn maps to hc m by a map of degree 2 which then maps to sh w furthermore we give a description of the degree 2 map in terms of rational holomorphic curves with constrained asymptotic markers in the symplectization of m

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

homotopy equivalence of nearby lagrangians and the serre spectral sequence we construct using relatively basic techniques a spectral sequence for exact lagrangians in cotangent bundles similar to the one constructed by fukaya seidel and smith that spectral sequence was used to prove that exact relative spin lagrangians in simply connected cotangent bundles with vanishing maslov class are homology equivalent to the base a similar result was also obtained by nadler the ideas in that paper were extended by abouzaid who proved that vanishing maslov class alone implies homotopy equivalence in this paper we present a short proof of the fact that any exact lagrangian with vanishing maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism when the fundamental group of the base is pro finite this implies homotopy equivalence

a cotangent fibre generates the fukaya category abstract we prove that the algebra of chains on the based loop space recovers the derived wrapped fukaya category of the cotangent bundle of a closed smooth oriented manifold the main new idea is the proof that a cotangent fibre generates the fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by cieliebak and latschev

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

predicates and predicate transformers for supervisory control of discrete event dynamical systems discrete event systems are studied treating the state space as the fundamental modeling concept the control of discrete event systems using predicates and predicate transformers is treated predicates have the advantage that they can concisely characterize an infinite state space the notion of controllability of a predicate is defined and the supervisory predicate control problem is solved a closed form expression for the weakest controllable predicate is obtained the problem of controlling discrete event systems under incomplete state observation is also considered and observability of predicates is defined techniques for finding extremal solutions of boolean equations are used to derive minimally restrictive supervisors

new obstructions to symplectic embeddings in this paper we establish new restrictions on the symplectic embeddings of basic shapes in symplectic vector spaces by refining an embedding technique due to guth we also show that they are sharp

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

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

the topology of kuranishi atlases kuranishi structures were introduced in the 1990s by fukaya and ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods starting from the same core idea patching local finite dimensional reductions we develop a theory of topological kuranishi atlases and cobordisms that transparently resolves algebraic and topological challenges in this virtual regularization approach it applies to any kuranishi type setting for example atlases with isotropy boundary and corners or lack of differentiable structure

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

cocolog a conditional observer and controller logic for finite machines the problem of observation and control for partially observed input state output machines is formulated in terms of a tree of first order logical theories a set of first order languages for the description of the controlled evolution and state estimation of any given machine cal m is specified further extralogical conditional control rules are formulated so that closed loop control actions occur when extralogically specified past observation dependent conditions are fulfilled in particular conditional control rules may include commands that steer the system state from a current partially observed state estimate to a target state if such a sequence of controls can be proven to exist starting from a general theory of cal m at the initial instant observations on the input output behaviour of the system at each later instant are accepted by the system as new axioms these are then used together with the previously generated theory to generate the current theory the acronym cocolog is used to denote the family of first order conditional observer and controller logics for any given input state output system a semantics is supplied for each cocolog system in terms of interpretations of controlled transitions on a tree indexed by the possible sequences of input output observations extralogical rules including the conditional control rules relating members of the family of theories of a cocolog system are presented in the form of a set of metalevel rules following the complete definition of a cocolog system the consistency and completeness of the first order theories in a cocolog system are established decidability is obtained using a unique model property and examples of the operation of a cocolog logic control system are given

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

transversality in elliptic morse theory for the symplectic action our goal in this paper is to settle some transversality question for the perturbed nonlinear cauchy riemann equations on the cylinder these results play a central role in the denition of symplectic floer homology and hence in the proof of the arnold conjecture there is currently no other reference to these transversality results in the open literature our approach does not require aronszajn s theorem instead we derive the unique continuation theorem from a generalization of the carleman similarity principle

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

the role of the legendre transform in the study of the floer complex of cotangent bundles consider a classical hamiltonian h on the cotangent bundle t m of a closed orientable manifold m and let l tm r be its legendre dual lagrangian in a previous paper we constructed an isomorphism from the morse complex of the lagrangian action functional that is associated to l to the floer complex that is determined by h in this paper we give an explicit construction of a homotopy inverse of contrary to other previously defined maps going in the same direction is an isomorphism at the chain level and preserves the action filtration its definition is based on counting floer trajectories on the negative half cylinder that on the boundary satisfy half of the hamilton equations albeit not of lagrangian type such a boundary condition defines fredholm operators with good compactness properties we also present a heuristic argument which independently of any fredholm and compactness analysis explains why the spaces of maps that are used in the definition of and are the natural ones the legendre transform plays a crucial role both in our rigorous and in our heuristic arguments we treat with some detail the delicate issue of orientations and show that the homology of the floer complex is isomorphic to the singular homology of the loop space of m with a system of local coefficients which is defined by the pullback of the second stiefel whitney class of tm on 2 tori in m 2015 wiley periodicals inc

obstruction bundles over moduli spaces with boundary and the action filtration in symplectic field theory branched covers of orbit cylinders are the basic examples of holomorphic curves studied in symplectic field theory since all curves with fredholm index one can never be regular for any choice of cylindrical almost complex structure we generalize the obstruction bundle technique of taubes for determining multiple cover contributions from gromov witten theory to the case of moduli spaces with boundary our result proves that the differential in rational symplectic field theory and contact homology is strictly decreasing with respect to the natural action filtration

gravitational descendants in symplectic field theory it was pointed out by y eliashberg in his icm 2006 plenary talk that the rich algebraic formalism of symplectic field theory leads to a natural appearance of quantum and classical integrable systems at least in the case when the contact manifold is the prequantization space of a symplectic manifold in this paper we generalize the definition of gravitational descendants in sft from circle bundles in the morse bott case to general contact manifolds after we have shown using the ideas in okounkov and pandharipande ann math 163 2 517 560 2006 that for the basic examples of holomorphic curves in sft that is branched covers of cylinders over closed reeb orbits the gravitational descendants have a geometric interpretation in terms of branching conditions we follow the ideas in cieliebak and latschev http arixiv org abs 0706 3284v2 math s6 2007 to compute the corresponding sequence of poisson commuting functions when the contact manifold is the unit cotangent bundle of a riemannian manifold

context free series on trees we investigate context free cf series on trees with coefficients on a semiring they are obtained as components of the least solutions of systems of equations having polynomials on their right hand sides the relationship between cf series on trees and cf tree grammars and recursive program schemes is also examined polypodes a new algebraic structure are introduced in order to study in common series on trees and words and applications are given

control of vector discrete event systems i the base model a vector discrete event system vdes is a discrete event system model in which the system state is represented by a vector with integer components and state transitions by integer vector addition the authors investigate such structures from the viewpoint of control theory the objective is to specialize to vdes the general automation based control theory of p j ramadge and w m wonham 1987 and w m wonham and p j ramadge 1988 it is shown that within the vdes framework a useful class of control systems can be compactly modeled and some of the associated control problems of ramadge and wonham can be more efficiently solved

lagrangian intersections and the serre spectral sequence for a transversal pair of closed lagrangian submanifolds l lof a sym plectic manifold m such that 1 l 1 l 0 c1 2 m 2 m and for a generic almost complex structure j we construct an invariant with a high homotopical content which consists in the pages of order 2o f as pec tral sequence whose differentials provide an algebraic measure of the high dimensional moduli spaces of pseudo holomorpic strips of finite energy that join l and l when l and lare hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the serre spectral sequence of the path loop fibration l pl l and we deduce some applications

the exponential decay of gluing maps for j holomorphic map moduli space abstract we prove the exponential decay of the derivative of the gluing maps with respect to the gluing parameter

corrigendum on the floer homology of cotangent bundles we fix an orientation issue that appears in our previous paper about the isomorphism between floer homology of cotangent bundles and loop space homology when the second stiefel whitney class of the underlying manifold does not vanish on 2 tori this isomorphism requires the use of a twisted version of the floer complex 2014 wiley periodicals inc

hasse diagrams for classes of deterministic bottom up tree to tree series transformations the relationship between classes of tree to tree series and o tree to tree series transformations computed by restricted deterministic bottom up weighted tree transducers is investigated essentially these transducers are deterministic bottom up tree series transducers except that the former are defined over monoids whereas the latter are defined over semirings and only use the multiplicative monoid thereof in particular the common restrictions of nondeletion linearity totality and homomorphism can equivalently be defined for deterministic bottom up weighted tree transducers using well known results of classical tree transducer theory and also new results on deterministic weighted tree transducers classes of tree to tree series and o tree to tree series transformations computed by restricted deterministic bottom up weighted tree transducers are ordered by set inclusion more precisely for every commutative monoid and all sensible combinations of the above mentioned restrictions the inclusion relation of the classes of tree to tree series and o tree to tree series transformations is completely conveyed by means of hasse diagrams

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

punctured holomorphic curves and lagrangian embeddings we use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and maslov class with boundary on lagrangian submanifolds of nonpositive curvature applications include the proof of audin s conjecture on the maslov class of lagrangian tori in linear symplectic space the construction of a new symplectic capacity obstructions to lagrangian embeddings into uniruled symplectic manifolds a quantitative version of arnold s chord conjecture and estimates on the size of weinstein neighbourhoods the main technical ingredient is transversality for the relevant moduli spaces of punctured holomorphic curves with tangency conditions

strongly fillable contact manifolds and j holomorphic foliations we prove that every strong symplectic filling of a planar contact manifold admits a symplectic lefschetz fibration over the disk and every strong filling of t 3 similarly admits a lefschetz fibration over the annulus it follows that strongly fillable planar contact structures are also stein fillable and all strong fillings of t 3 are equivalent up to symplectic deformation and blowup these constructions result from a compactness theorem for punctured j holomorphic curves that foliate a convex symplectic manifold we use it also to show that the compact v supported symplectomorphism group on t t 2 is contractible and to define an obstruction to strong fillability that yields a non gauge theoretic proof of gay s recent nonfillability result g for contact manifolds with positive giroux torsion

morse bott homology we give a new proof of the morse homology theorem by constructing a chain complex associated to a morse bott smale function that reduces to the morse smale witten chain complex when the function is morse smale and to the chain complex of smooth singular n cube chains when the function is constant we show that the homology of the chain complex is independent of the morse bott smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a floer type continuation theorem

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

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

virtual moduli cycles and gromov witten invariants of noncompact symplectic manifolds this paper constructs and studies the gromov witten invariants and their properties for noncompact geometrically bounded symplectic manifolds two localization formulas for gw invariants are also proposed and proved as applications we get solutions of the generalized string equation and dilation equation and their variants the more solutions of wdvv equation and quantum products on cohomology groups are also obtained for the symplectic manifolds with finitely dimensional cohomology groups to realize these purposes we further develop the language introduced by liu tian to describe the virtual moduli cycle defined by liu tian fukaya ono li tian ruan and siebert

recognizable formal power series on trees trees are a very basic object in computer science they intervene in nearly a ly domain and they are studied for their own or used to represent conveniently a given situation there are at least three directions where investigations on trees themselves are motivated and this for different reasons first the notion of tree is the basis of algebraic semantics nivat 19 rosen 2 2 etc in this context the study of special languages of trees i e forests their classification and their behaviour under various types of transformations are of great importance arnold fl dauchet 8 j l ilin 1161 mongy 18 by essence work in this area is an extension of the algebraic theory of languages trees and ianguages are in fact directly related via the derivation trees of an algebraic grammar thatcher 25 a second topic heavily related to trees concerns dzra structures trees mainly binary trees and its variants constitute one of the most widely known data structures see e g hlnuth 15 the analysis of the worst cale expected or average running time t ehzvgrr nfi certain algorithms requires sometimes long and delicate computations flajolet lo kemp 14 flajolet and steyaert 12 finally trees occupy a distinguished place in the enumeration of graphs and maps both because of the simplicity of their structure and for the relationship between their el lcodings and aigebraic languages the nature of the enumerating series and especially the question whether they are algebraic or not is one of the central problems in this domain cori 7 chottin 4 we propose here a theory of formal power series on trees and present some of their basic properties together with various examples of applications which as we hope will show the interest of its development within the framework we just sketched a formal power series on trees is a function which gssociates a number to esch tree thus we could also have called them tree functions ii in analogy with the term word function used by several authors paz and salomaa zo cobham 5 as an equivalent denomination for formal power series on words the main goal of a formal power series is to count or to represent the result of some computation on

examples of symplectic structures in this paper we construct symplectic forms tilde omega k k geqq 0 on a compact manifold tilde y which have the same homotopy theoretic invariants but which are not diffeomorphic

contact homology and virtual fundamental cycles we give a construction of contact homology in the sense of eliashberg givental hofer specifically we construct coherent virtual fundamental cycles on the relevant compactified moduli spaces of pseudo holomorphic curves the aim of this work is to provide a rigorous construction of contact homology an invariant of contact manifolds and exact symplectic cobordisms due to eliashberg givental hofer eli98 egh00 the contact homology of y is defined in terms of pseudo holomorphic curve counts in the sense of gromov gro85 in the symplectization r y unfortunately the moduli spaces of such curves often suffer from a severe lack of transversality and for this reason it has until now remained an important open problem to define the relevant curve counts in full generality we will use the framework developed in par15 to construct coherent virtual fundamental cycles on the moduli spaces in question thus giving rise to the desired curve counts our methods are quite general and should apply equally well to many other moduli spaces of interest previously a number of important special cases of contact homology and closely related invariants have been constructed rigorously using generic and or automatic transversality techniques cylindrical contact homology of some three manifolds was constructed by bao

the degenerate special lagrangian equation abstract this article introduces the degenerate special lagrangian equation dsl and develops the basic analytic tools to construct and study its solutions the dsl governs geodesics in the space of positive graph lagrangians in c n existence of geodesics in the space of positive lagrangians is an important step in a program for proving existence and uniqueness of special lagrangians moreover it would imply certain cases of the strong arnold conjecture from hamiltonian dynamics we show the dsl is degenerate elliptic we introduce a space time lagrangian angle for one parameter families of graph lagrangians and construct its regularized lift the superlevel sets of the regularized lift define subequations for the dsl in the sense of harvey lawson we extend the existence theory of harvey lawson for subequations to the setting of domains with corners and thus obtain solutions to the dirichlet problem for the dsl in all branches moreover we introduce the calibration measure which plays a role similar to that of the monge ampere measure in convex and complex geometry the existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the dsl have a well defined length in the space of positive lagrangians

coherent orientations in symplectic field theory we study the coherent orientations of the moduli spaces of holomorphic curves in symplectic field theory generalizing a construction due to floer and hofer in particular we examine their behavior at multiple closed reeb orbits under change of the asymptotic direction the orientations are determined by a certain choice of orientation at each closed reeb orbit that is similar to the orientation of the unstable tangent spaces of critical points in finite dimensional morse theory

heegaard floer homology of broken fibrations over the circle abstract we extend perutz s lagrangian matching invariants to 3 manifolds which are not necessarily fibered using the technology of holomorphic quilts we prove an isomorphism of these invariants with ozsvath szabo s heegaard floer invariants for certain extremal spin c structures as applications we give new calculations of heegaard floer homology of certain classes of 3 manifolds and a characterization of juhasz s sutured floer homology

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 homology for the heat flow we use the heat flow on the loop space of a closed riemannian manifold viewed as a parabolic boundary value problem for infinite cylinders to construct an algebraic chain complex the chain groups are generated by perturbed closed geodesics the boundary operator is defined by counting modulo time shift heat flow trajectories between geodesics of morse index difference one by salamon and weber gafa 16 1050 138 2006 this heat flow homology is naturally isomorphic to floer homology of the cotangent bundle for hamiltonians given by kinetic plus potential energy

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

a temporal logic approach to discrete event control for the safety canonical class this paper presents a temporal logic formulation of discrete event control which forms a new theoretical basis for control analysis and synthesis of a class of discrete event systems des based on the formulation a basic supervisory control theory is developed for a control objective specified by an invariance formula belonging to the safety canonical class of manna and pneuli using the safety canonical class as a basis the refinement and generalization of the existing basic predicate framework are demonstrated a simple example illustrates the formal axiomatic means to perform control theoretic analysis and synthesis under the new formulation