number of orbits of branch points of r trees an r tree is a metric space in which any two points are joined by a unique arc every arc is isometric to a closed interval of r when a group g acts on a tree z tree x without inversion then x g is a graph one gets a presentation of g from the quotient graph x g with vertex and edge stabilizers attached for a general r tree x there is no such combinatorial structure on x g but we can still ask what the maximum number of orbits of branch points of free actions on r trees is we prove the finiteness of the maximum number for a family of groups which contains free products of free abelian groups and surface groups and this family is closed under taking free products with amalgamation

algebraic convergence of schottky groups a discrete faithful representation of the free group on g generators f g into isom h 3 is said to be a schottky group if h 3 d is homeomorphic to a handlebody h g where d is the domain of discontinuity for s action on the sphere at infinity for h 3 schottky space s g the space of all schottky groups is parameterized by the quotient of the teichmuller space f s g of the closed surface of genus g by mod 0 h g where mod 0 h g is the group of isotopy classes of homeomorphisms of s g which extend to homeomorphisms of h g which are homotopic to the identity masur exhibited a domain o h g of discontinuity for mod 0 h g s action on pl s g the space of projective measured laminations on s g so b h g o h g mod 0 h g may be appended to s g as a boundary

instability of high dimensional hamiltonian systems multiple resonances do not impede diffusion abstract we consider models given by hamiltonians of the form h i p q t e h i j 1 n 1 2 p j 2 v j q j e q i p q t e where i i r d t d p q r n t t 1 these are higher dimensional analogues both in the center and hyperbolic directions of the models studied in 28 29 43 and are usually called a priori unstable hamiltonian systems all these models present the large gap problem we show that for 0 e 1 under regularity and explicit non degeneracy conditions on the model there are orbits whose action variables i perform rather arbitrary excursions in a domain of size o 1 this domain includes resonance lines and hence large gaps among d dimensional kam tori this phenomenon is known as arnold diffusion the method of proof follows closely the strategy of 28 29 the main new phenomenon that appears when the dimension d of the center directions is larger than one is the existence of multiple resonances in the space of actions i i r d we show that since these multiple resonances happen in sets of codimension greater than one in the space of actions i they can be contoured this corresponds to the mechanism called diffusion across resonances in the physics literature the present paper however differs substantially from 28 29 on the technical details of the proofs we have taken advantage of the theory of the scattering map developed in 31 notably the symplectic properties which were not available when the above papers were written we have analyzed the conditions imposed on the resonances in more detail more precisely we have found that there is a simple condition on the melnikov potential which allows us to conclude that the resonances are crossed in particular this condition does not depend on the resonances so that the results are new even when applied to the models in 28 29

transfer of energy to high frequencies in the cubic defocusing nonlinear schrodinger equation we consider the cubic defocusing nonlinear schrodinger equa tion on the two dimensional torus we exhibit smooth solutions for which the support of the conserved energy moves to higher fourier modes this behavior is quantified by the growth of higher sobolev norms given any 1 k 1 s 1 we construct smooth initial data u0 withu0 h s k at some time t this growth occurs despite the hamiltonian s bound onu t h 1 and despite the conservation of the quantityu t l2

arnold diffusion in arbitrary degrees of freedom and normally hyperbolic invariant cylinders we prove a form of arnold diffusion in the a priori stable case let h 0 p epsilon h 1 theta p t quad theta in mathbb t n p in b n t in mathbb t mathbb r mathbb t be a nearly integrable system of arbitrary degrees of freedom n geqslant 2 with a strictly convex h0 we show that for a generic epsilon h 1 there exists an orbit theta p satisfying p t p 0 l h 1 0 where l h 1 is independent of epsilon the diffusion orbit travels along a codimension 1 resonance and the only obstruction to our construction is a finite set of additional resonances

stable actions of groups on real trees this paper further develops rips s work on real trees we study a class of actions called stable which includes actions with trivial arc stabilizers and small actions of hyperbolic groups

exact image theory for the sommerfeld half space problem part iii general formulation the general sommerfeld problem with both epsilon and mu discontinuous and a source consisting of arbitrarily oriented electric and or magnetic dipoles at the same location is considered in terms of image theory the problem is formulated with electric and magnetic fields instead of potential quantities resulting in a vector transmission line interpretation of the fourier transformed problem the image sources are seen to be located in complex space expressable in terms of a certain basic image current function which was encountered in part ii of this paper on the vertical electric dipole problem the horizontal electric magnetic dipole image is solved and found to consist of both vertical and horizontal current components the image concept is generalized to the most general three dimensional sources as a check the well known reflection coefficient method is obtained as the far field approximation of the present theory

chaos in the gylden problem we consider the gylden problem a perturbation of the kepler problem via an explicit function of time for certain general classes of planar periodic perturbations after proving a poincare melnikov type criterion we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi infinite sequences with infinitely many symbols we achieve the main result by computing the melnikov integral explicitly

construction of invariant whiskered tori by a parameterization method part i maps and flows in finite dimensions abstract we present theorems which provide the existence of invariant whiskered tori in finite dimensional exact symplectic maps and flows the method is based on the study of a functional equation expressing that there is an invariant torus we show that given an approximate solution of the invariance equation which satisfies some non degeneracy conditions there is a true solution nearby we call this an a posteriori approach the proof of the main theorems is based on an iterative method to solve the functional equation the theorems do not assume that the system is close to integrable nor that it is written in action angle variables hence we can deal in a unified way with primary and secondary tori it also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant linear map the a posteriori formulation allows us to justify approximate solutions produced by many non rigorous methods e g formal series expansions numerical methods the iterative method is not based on transformation theory but rather on successive corrections this makes it possible to adapt the method almost verbatim to several infinite dimensional situations which we will discuss in a forthcoming paper we also note that the method leads to fast and efficient algorithms we plan to develop these improvements in forthcoming papers

regularity of harmonic maps to trees we derive regularity theorems and local finiteness results in this paper we first prove the regularity type theorem for harmonic maps to r trees that is when the order of a harmonic map at a point is sufficiently close to 1 then this point is a regular point we then prove a regularity theorem near higher order points we show that if the image of a fixed ball of a properly normalized harmonic map is sufficiently close to a subtree then the image of a smaller ball of this map lies in that subtree as an application we prove that the local image of a harmonic map to an r tree lies in a finite homogeneous subtree

efficient evaluation of the finite part of pole free sommerfeld integrals in half space problems with predefined accuracy this communication presents a novel simple and robust approach for the computation of the finite part of pole free sommerfeld integrals sis in half space problems with high and controllable accuracy over a large range of source observer distances the approach includes the following techniques 1 cancellation of the branch point singularities based on the square root change of variables for numerical integration 2 approximation of real axis integration path in order to enhance the singularity cancellation for arbitrary low loss dielectrics 3 thresholds for truncation of the interval of integration for given accuracy which improve the efficiency of computation and 4 prediction formulas that estimate the required number of integration points for a given accuracy up to 1000 wavelengths of source observer distance the proposed approach is verified through numerical examples and comparison to reference methods

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

efficient calculation of the fields of a dipole radiating above an impedance surface in this paper the classic problem of field computation for an infinitesimal dipole radiating above an impedance half space is revisited the expressions for the traditional solution consist of integrals of the sommerfeld type that cannot be evaluated in closed form and due to their highly oscillatory nature are difficult to evaluate numerically a method known as exact image theory which has previously been applied to vertical electric and magnetic dipoles is used to derive explicit expressions for dipoles of arbitrary orientation above impedance surfaces starting from the spectral representation of the field the reflection coefficients are cast in the form of exact laplace transforms and then by changing the order of integrations field expressions in terms of rapidly converging integrals are obtained these expressions are exact and valid for any arbitrary source alignment or observation position it is shown that the formulation for a horizontal dipole contains an image in the conjugate complex plane resulting in a diverging exponential term not previously addressed in the literature it is shown through further mathematical manipulations that the diverging term is a contribution of the mirror image which can be extracted comparison of numerical results from exact image theory and the original sommerfeld type expressions shows good agreement as well as a speedup in computation time of many orders of magnitude which depends on the distance between the transmitter and the receiver this formulation can effectively replace the approximate asymptotic expressions used for predicting wave propagation over a smooth planar ground having different regions of validity the exact image formulation is also of practical use in evaluation of the green s function for various applications in scattering problems where approximate solutions are not sufficient

taylor expansion based fast multipole method for 3 d helmholtz equations in layered media abstract in this paper we develop a taylor expansion te based fast multipole method fmm for low frequency 3d helmholtz green s function in layered media two forms of taylor expansions with either non symmetric or symmetric derivatives of layered media green s functions are used for the implementations of the proposed te fmm in the implementation with non symmetric derivatives an algorithm based on discrete complex image approximations and recurrence formulas is shown to be very efficient and accurate in computing the high order derivatives meanwhile the implementation based on symmetric derivatives is more robust and pre computed tables for the high order derivatives in translation operators are used numerical tests in layered media have validated the accuracy and o n complexity of the proposed algorithms

hyperbolic structures on 3 manifolds i deformation of acylindrical manifolds this is the first in a series of papers showing that haken manifolds have hyperbolic structures this first was published the second two have existed only in preprint form and later preprints were never completed this eprint is only an approximation to the published version which is the definitive form for part i and is provided for convenience only all references and quotations should be taken from the published version since the theorem numbering is different and not all corrections have been incorporated into the present version parts ii and iii will be made available as eprints shortly

fast multipole representation of green s function for an impedance half space in this paper exact and approximate multipole representations for fields of a horizontal or a vertical infinitesimal dipole above impedance surfaces are derived from the exact image formulation these multipole expansions are appropriate for applications in the method of moments through a formulation known as the fast multipole method unlike the existing approximate formulations these new multipole representations of the green s function for the half space impedance surface problem do not impose any restrictions on the location of source or observation points and are computationally very efficient

degenerations of hyperbolic structures ii measured laminations in 3 manifolds that paper concerned the general theory of groups acting on r trees and the relationship of these actions to representations into sl2 c the purpose of the present paper is to develop the foundations of a general theory of measured laminations in 3 manifolds in the third paper 8 the results of this paper will be applied to study actions of 3 manifold groups on trees the starting point of 8 is the idea that dual to an action of qg m on an r tree is a codimension 1 measured lamination in m combining the latter results with 7 gives information about how hyperbolic structures on 3 manifolds can degenerate we view measured laminations in 3 manifolds as generalizations of both geodesic laminations on surfaces and surfaces in 3 manifolds most of this paper consists in extending thurston s theory of geodesic laminations and the hakenstallings waldhausen theory of incompressible surfaces to our context the paper is organized along the following lines in chapter i we develop the basic notions of codimension 1 measured laminations we prove one new and quite useful result theorem i 3 2 concerning a decomposition of these objects we also give a definition of the euler characteristic of a measured lamination and show that it has all the usual properties in chapter ii we introduce the branched surfaces of williams 16 and hatcher thurston we establish analogues for measured laminations carried by branched surfaces of some results of thurston s for train tracks and geodesic laminations by imitating an argument of plante s 10 we show that the leaves of a measured lamination have polynomial growth this is used to show that if a branched surface n carries some lamination all of whose leaves have virtually abelian fundamental groups and carries no disks or spheres then all laminations carried by n have zero euler characteristic we express the latter property by saying that n is flat in chapter iii generalizing haken s notion of normal surfaces we develop the theory of normal measured laminations and of normal branched surfaces in a

infinitely many hyperbolic 3 manifolds which contain no reebless foliation calegari ca has shown that if m is also atoroidal then or1 m is gromov negatively curved furthermore thurston has proposed an approach to demonstrating geometrization for such m many 3 manifolds contain reebless foliations and it has often been conjectured that all closed hyperbolic 3 manifolds do it is our impression that for many years hatcher provided the sole voice of dissent in this paper we give the first examples of closed hyperbolic 3 manifolds which contain no reebless foliation

exact closed form expression of the electromagnetic field excited by a uniform current distribution lying on a cartesian quadrant the exact analytical solution of the electromagnetic field distribution produced by uniform electric currents excited on cartesian half planes and quadrants is presented the total field is expressed in terms of geometrical optics go and diffracted field contributions that remain valid for arbitrary observation points and frequency the jump discontinuity of the go field is exactly compensated by the diffracted field whose spatial distribution is described in terms of the incomplete hankel functions and by means of a novel special function the expression of the diffracted field includes contributions arising from the edges and the vertex of the considered cartesian domain illustrating the analytical behavior of the near field singularities and providing insight into the physical mechanisms governing the field diffractive processes the proposed solution yields a method to determine the physical optics po response of flat metallic screens excited by uniform plane waves as well as the fields produced by pulse shaped basis functions used in the method of moments mom solution of electromagnetic problems showing the relevance of the exact analytical expressions of the go and diffracted fields arising from edges and vertices of each rectangular pulse domain numerical examples validate the accuracy of the proposed field representation

limits of geometrically tame kleinian groups let f be a finitely generated geometrically tame kleinian group with h3 f m when m has a finite volume the hyperbolic structure of m is unique up to homotopy by mostow s rigidity theorem but when m has infinite volume it is meaningful to consider the deformation space of the hyperbolic structure of m let ah m denote the space of faithful representations with discrete image from nl m to psl2 c modulo conjugacy with the topology of algebraic convergence until now only a little is known about ah m when f is of second kind there exists the space of quasi conformal deformation of f denoted by qho m with the quasi conformal or quasi isometric topology there is a canonical continuous injection t qho m ah m we abuse the notation qho m for t qho m the topological type of qho m has been determined by sullivan 21 one method for investigating ah m is to consider limits of sequences in qho m boundary groups in bers 3 and double limits in thurston 24 are such examples for considering limits it is important to determine when a sequence in ah m converges it was shown by thurston 24 and morgan shalen 16 that when m is acylindrical every sequence in ah m has a converging subsequence we gave in ohshika 19 a sufficient condition of convergence for geometrically finite groups in w 2 of this paper we prove a theorem which implies the existence of converging subsequence in all of the above cases the case of boundary groups double limits acylindrical manifolds and the theorem in 19 the statement is as follows

incomplete hankel and modified bessel functions a class of special functions for electromagnetics a novel class of special functions for electromagnetics is presented formed by the incomplete hankel and modified bessel functions this class allows solving electromagnetics problems concerning truncated cylindrical structures the differential and recurrence equations of these functions feature additional terms with respect to the classical theory of the hankel and bessel functions the general properties the most important analytical characteristics and the large argument asymptotic approximations of the incomplete functions are derived using the steepest descent path sdp technique showing that each special function splits into two terms the first one has a discontinuous character and is linked to the saddle point s contribution s while the second one arising from the integral end point contribution s compensates exactly the said discontinuity in the solution of electromagnetic problems the first term describes the geometrical optics go field the diffracted field being described by the second one the general theory is employed to find the closed form analytical solution of the field radiated from a uniform line current source using the properties of the incomplete hankel functions it is demonstrated that this source excites cylindrical fields having optical character finally the shape of the spatial regions where the go solution cannot be applied is determined and discussed in details

transition tori in the five body problem abstract transition tori are invariant tori connected by transverse heteroclinic orbits as shown by arnold nearby orbits can drift from torus to torus in this paper a family of transition tori is constructed for the planar five body problem

exact closed form expression of a sommerfeld integral for the impedance plane problem a sommerfeld integral for an impedance half plane is one of the classical problems in electromagnetic theory in this paper the integral is evaluated into two series representations which are expressed in terms of the exponential integral and the lommel function respectively it is shown that the series expansions absolutely converge for any parameters such as distance or surface impedance then based on the lommel function expansion an exact closed form expression of the integral is formulated this expression is written in terms of incomplete weber integrals which are directly related to incomplete cylindrical functions such as incomplete lipschitz hankel integrals additionally a complete asymptotic series of the integral is obtained based on the exponential integral expansion with conventional methods such as steepest descent method it is cumbersome to derive the divergent series the validity of all the formulations derived in this paper is demonstrated through a comparison with a numerical integration of the integral for various situations

essential laminations in 3 manifolds it is a well established principle in manifold topology that by studying codimension 1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself this is particularly true for 3 dimensional manifolds using incompressible surfaces a great many advances have been made during the last 30 years the most spectacular results were obtained by haken ha waldhausen w and thurston t1 more recently work t2 g14 using taut foliations in 3 manifolds has proven fruitful unfortunately in some sense most closed 3 manifolds do not contain incompressible surfaces and it is currently not known exactly which 3 manifolds have taut foliations the purpose of this paper is to study another codimension 1 object the essential lamination which is a generalization of the incompressible surface as well as the taut foliation we will show that the existence of an essential lamination in a 3 manifold m implies that m has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation we will also show that the universal cover of a manifold containing an essential lamination is r3 precise definitions of terms used in the following theorem will be given in section 1 here we give only a rough idea we begin with two alternative approximate definitions of essential lamination i an essential lamination in a 3 manifold m is a lamination satisfying four conditions the inclusion of leaves of the lamination into m should induce an injection on stj the complement of the lamination should be irreducible no leaf should be a sphere and the lamination should be end incompressible to say that a lamination is end incompressible means roughly that a folded leaf can be straightened using an isotopy there are no infinite folds ii alternatively

exact image theory for the sommerfeld half space problem part ii vertical electrical dipole applying the laplace transform the exact distributed image current function is obtained for the classical sommerfeld half space problem with vertical magnetic current source the resulting field integral is well behaved when the image current is situated in complex space unlike previous approximate images the present theory is valid for any distance height of the source frequency and half space parameters it is demonstrated that the present image theory reduces to the well known dipole image at complex depth for large dielectric parameters of the half space also the reflection coefficient method is obtained as a farfield approximation calculation of fields through exact image integration is seen to be simple and accurate and require modest computer capacity and time in an appendix some properties of the multivalued green s function arising from a dipole source in complex space are also studied

extrapolation methods for sommerfeld integral tails a review is presented of the extrapolation methods for accelerating the convergence of sommerfeld type integrals i e semi infinite range integrals with bessel function kernels which arise in problems involving antennas or scatterers embedded in planar multilayered media attention is limited to partition extrapolation procedures in which the sommerfeld integral is evaluated as a sum of a series of partial integrals over finite subintervals and is accelerated by an extrapolation method applied over the real axis tail segment spl alpha spl infin of the integration path where spl alpha 0 is selected to ensure that the integrand is well behaved an analytical form of the asymptotic truncation error or the remainder which characterizes the convergence properties of the sequence of partial sums and serves as a basis for some of the most efficient extrapolation methods is derived several extrapolation algorithms deemed to be the most suitable for the sommerfeld integrals are described and their performance is compared it is demonstrated that the performance of these methods is strongly affected by the horizontal displacement of the source and field points spl rho and by the choice of the subinterval break points furthermore it is found that some well known extrapolation techniques may fail for a number of values of spl rho and ways to remedy this are suggested finally the most effective extrapolation methods for accelerating sommerfeld integral tails are recommended

orbits of unbounded energy in quasi periodic perturbations of geodesic flows abstract we show that certain mechanical systems including a geodesic flow in any dimension plus a quasi periodic perturbation by a potential have orbits of unbounded energy the assumptions we make in the case of geodesic flows are a the metric and the external perturbation are smooth enough b the geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection c the frequency of the external perturbation is diophantine d the external potential satisfies a generic condition depending on the periodic orbit considered in b the assumptions on the metric are c 2 open and are known to be dense on many manifolds the assumptions on the potential fail only in infinite codimension spaces of potentials the proof is based on geometric considerations of invariant manifolds and their intersections the main tools include the scattering map of normally hyperbolic invariant manifolds as well as standard perturbation theories averaging kam and melnikov techniques we do not need to assume that the metric is riemannian and we obtain results for finsler or lorentz metrics indeed there is a formulation for hamiltonian systems satisfying scaling hypotheses we do not need to make assumptions on the global topology of the manifold nor on its dimension

complete closed form expression of dyadic green s function and its far and near field approximations for an impedance half plane a closed form expression of the dyadic green s function is formulated for an impedance half plane which is written in terms of the incomplete cylindrical function of poisson form due to the branch cut of the logarithm function that is required to calculate the input argument of the incomplete cylindrical function the closed form representation consists of two formulations since the closed form expression contains a singularity at 0 the small argument expansion of the expression is also derived to rigorously characterize the behavior of the function at 0 the previously reported complete asymptotic expansion for the sommerfeld integral for an impedance half plane is not accurate for practically important cases such as near earth propagation and or when the surface is highly conductive hence in this paper a new complete asymptotic series of the sommerfeld integral are derived for the case that the existing asymptotic series is not accurate the two asymptotic series not only allow efficient numerical computation but also provide more accurate results for virtually all propagation scenarios based on the two asymptotic series the complete asymptotic series of the dyadic green s function is derived all derived formulations are numerically verified and their accuracies are investigated

the nielsen thurston classification and automorphisms of a free group i using train tracks and invariant laminations thurston has developed a whole theory to understand the dynamics and geometry of diffeomorphisms of surfaces th ca bl by introducing a combinatorial analogue of train tracks m bestvina and m handel be ha have managed to analyze irreducible automorphisms of a free group and using this analysis to bound the rank of the fixed subgroup of an automorphism by the rank of the ambient group which was known before as the scott conjecture in se1 we introduced a dynamical algebraic commutative diagram associated with a limit action of the ambient free group fn on some real tree this commutative diagram allows one to interpret dynamical invariants of the limit action in terms of algebraic properties of the automorphism in question and vice versa using this diagram we showed how to obtain the nielsen thurston classification of automorphisms of surfaces on the algebraic level and some generalized versions of the scott conjecture in this paper we make an extensive use of the dynamical algebraic commutative diagram and the classification of stable actions of finitely presented groups on real trees ri be fe1 to construct a canonical hierarchical decomposition of a free group associated with an automorphism of it which is from our point of view the analogue of the nielsen thurston classification for automorphisms of free groups it is also our belief that our hierarchical decomposition should serve as a complementary to the bestvina handel train tracks and invariant laminations in studying the dynamics and combinatorics of general rather than irreducible automorphisms of a free group