the topology of deformation spaces of kleinian groups let m be a compact hyperbolizable 3 manifold with nonempty incompressible boundary and let ah 1 m denote the space of conjugacy classes of discrete faithful representations of 1 m i nto psl 2 c the components of the interior mp 1 m of ah 1 m as a subset of the appropriate representation variety are enumerated by the spacea m of marked homeomorphism types of oriented compact irreducible 3 manifolds homotopy equivalent to m in this paper we give a topological enumeration of the components of the closure of mp 1 m and hence a conjectural topological enumeration of the components of ah 1 m we do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable kleinian groups we use this enumeration to exhibit manifolds m for which ah 1 m has inflnitely many components in this paper we begin a study of the global topology of deformation spaces of kleinian groups the basic object of study is the space ah 1 m of marked hyperbolic 3 manifolds homotopy equivalent to a flxed compact 3manifold m the interior mp 1 m of ah 1 m is very well understood due to work of ahlfors bers kra marden maskit sullivan and thurston in particular the components ofmp 1 m are enumerated by topological data namely the set a m of marked compact oriented irreducible 3 manifolds homotopy equivalent to m while each component is parametrized by analytic data coming from the conformal boundaries of the hyperbolic 3 manifolds thurston s ending lamination conjecture provides a conjectural classiflcation for elements of ah 1 m by data which are partially topologi

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

translation equivalence in free groups motivated by the work of leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces we investigate a similar phenomenon for free groups namely we study the situation when two elements g h in a free group f have the property that for every free isometric action of f on an r tree x the translation lengths of g and h on x are equal

author s reply to comments on nonlinear repetitive control our paper ghosh and paden nonlinear repetitive control ieee trans automat contr vol 45 p 949 54 2000 proves theorem 1 which establishes conditions for the stability of the nonlinear repetitive control system and proposition 2 which goes on to establish bounds on the steady state error signals these are correct as stated as they apply to repetitive controllers with a finite number of oscillators in their internal model an issue correctly raised in the commentary lucibello comments on nonlinear repetitive control ieee trans automat contr vol 48 p 1470 1 2003 is that the conditions of proposition 2 might be difficult to meet when the number of oscillators is large this observation does not challenge the veracity of the proposition but is certainly insightful in our introductory comments of section 11 we attempt and fail to make a qualitative connection to the pure time delay type repetitive control the following statement is incorrect and should be deleted from our paper this justifies the fact that in the limit n spl rarr spl infin the above model of the repetitive controller works as well as the ideal model in achieving perfect tracking for typical reference signals

lifting representations to covering groups the primary purpose of this note is to prove that if a discrete subgroup z of z s c has no 2 torsion then it lifts to s c i e there is a homomorphism z s c such that the composition with the natural projection s c z s c is the identity on k note that the absence of 2 torsion is necessary since cyclic subgroups of even order do not lift this extends a result of km 2 for which z was required to be a function group our methods are simple and straightforward requiring only elementary covering space theory

a family of non injective skinning maps with critical points certain classes of 3 manifolds following thurston give rise to a skinning map a self map of the teichmuller space of the boundary this paper examines the skinning map of a 3 manifold m a genus 2 handlebody with two rank 1 cusps we exploit an orientation reversing isometry of m to conclude that the skinning map associated to m sends a specified path to itself and use estimates on extremal length functions to show non monotonicity and the existence of a critical point a family of finite covers of m produces examples of non immersion skinning maps on the teichmuller spaces of surfaces in each even genus and with either 4 or 6 punctures

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

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

monopoles and lens space surgeries monopole floer homology is used to prove that real projective three space cannot be obtained from dehn surgery on a nontrivial knot in the three sphere to obtain this result we use a surgery long exact sequence for monopole floer homology together with a nonvanishing theorem which shows that monopole floer homology detects the unknot in addition we apply these techniques to give information about knots which admit lens space surgeries and to exhibit families of three manifolds which do not admit taut foliations

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

taut foliations in punctured surface bundles i this model a given fibration can be perturbed to give nearby fibrations in this paper we describe an alternative way of perturbing fibrations with non empty boundary instead of fibrations this construction yields taut foliations very closely related to the original fibration but with different boundary behaviour given m with torus boundary component t and assigned coordinate system for t we say that a fibration or taut foliation f realizes slope r on t iff intersects t in simple curves of slope r when f is a fibration r is necessarily rational in general however we assume only that r is real when a compact orientable fibred manifold m has a single boundary component exactly one slope is realized by fibrations however the closely related taut foliations realize a non empty interval of slopes in 4 we prove the following theorem theorem 4 1 given an orientable fibred compact 3 manifold with single boundary component a fibration with fibre surface of negative euler charac teristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration remark when m is fibred with fibre a once punctured torus the taut foliations realized are those of 11 in different guise as corollary we have information about manifolds obtained by performing dehn surgery along slopes sufficiently close to the slope of a fibre such manifolds necessarily contain taut foliations hence by novikov 16 they are irreducible with infinite fundamental group by palmeira 20 they have universal cover r 3 in certain cases it is possible to say precisely what sufficiently close means that is to identify the interval realized in 11 hatcher does this for the once punctured torus bundles over s 1 i n 23 we describe the intervals for once punctured surface bundles in terms of their gabai degeneracy in 5 we begin an investigation of generalizations of the construction we give an analogous result for manifolds containing a taut foliation with compact leaf a punctured surface of positive genus and possessing a non separating comple mentary product disc see 5 then for those readers familiar with essential laminations we show that if m contains an essential spanning surface s with non separating complementary product disc then s can be perturbed to yield

plissage des varietes hyperboliques de dimension 3 we give a characterization of the measured geodesic laminations which can occur as the bending measured lamination of some geometrically finite metric on a 3 manifold when the 3 manifold has incompressible boundary such a characterization has already been given by f bonahon and j p otal here we deal with the general case

perturbation theory of a symmetric center within lienard equations in this article we introduce the use of lambert function to develop further the global perturbation theory of an integrable lienard equation which displays a symmetric center we prove a global morse lemma for the first integral and deduce the existence of an associated picard fuchs system we revisit previous contributions to first order perturbation theory with the help of these new analytic techniques and in particular we check that the fundamental integrals are linearly independent the lambert function allows to find an expansion formula for these integrals we also study the possibility to develop a higher order perturbation theory the algorithm of the successive derivatives works in general in the class of analytic functions on the domain d where the level sets of the first integral are ovals we end the article with some results on the first integral of a symmetric lienard equation deduced from the algorithm of successive derivatives

non realizability and ending laminations proof of the density conjecture we give a complete proof of the bers sullivan thurston density conjecture in the light of the ending lamination theorem it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds the ending invariants are either riemann surfaces or filling laminations supporting masur domain measured laminations and satisfying some mild additional conditions with any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence we derive the necessary compactness theorem combining the rips machine with non existence results for certain small actions on real trees of free products of surface groups and free groups we prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface in order to be able to apply the ending lamination theorem we have to prove that this algebraic limit has the desired topological type and that these non realized laminations are ending laminations that this is the case is the main novel technical result of this paper loosely speaking we show that a filling masur domain lamination which is not realized is an ending lamination

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

laminations and groups of homeomorphisms of the circle if m is an atoroidal 3 manifold with a taut foliation thurston showed that 1 m acts on a circle here we show that some other classes of essential laminations also give rise to actions on circles in particular we show this for tight essential laminations with solid torus guts we also show that pseudo anosov flows induce actions on circles in all cases these actions can be made into faithful ones so 1 m is isomorphic to a subgroup of homeo s1 in addition we show that the fundamental group of the weeks manifold has no faithful action on s1 as a corollary the weeks manifold does not admit a tight essential lamination with solid torus guts a pseudo anosov flow or a taut foliation finally we give a proof of thurston s universal circle theorem for taut foliations based on a new purely topological proof of the leaf pocket theorem

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

new family of abelian integrals satisfying chebyshev property abstract this paper is motivated by the chebyshev property of a family of abelian integrals along non algebraic level ovals a new criterion is established to study the chebyshev property of these abelian integrals as its applications we give three examples to show that the ordered sets defined by abelian integrals are ect systems moreover the number of the abelian integrals can be an arbitrary positive integer

heegaard surfaces and measured laminations i the waldhausen conjecture we give a proof of the so called generalized waldhausen conjecture which says that an orientable irreducible atoroidal 3 manifold has only finitely many heegaard splittings in each genus up to isotopy jaco and rubinstein have announced a proof of this conjecture using different methods

promoting essential laminations we show that a co orientable taut foliation of a closed orientable algebraically atoroidal 3 manifold is either the weak stable foliation of an anosov flow or else there are a pair of very full genuine laminations transverse to the foliation

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

ubiquity of geometric finiteness in boundaries of deformation spaces of hyperbolic 3 manifolds we show that geometrically finite kleinian groups are dense in the boundary of the quasiconformal deformation space of any geometrically finite kleinian group

projective structures with degenerate holonomy and the bers density conjecture we prove the bers density conjecture for singly degenerate kleinian surface groups without parabolics

stabilization of approximately feedback linearizable systems using singular perturbation we consider a stabilization problem of approximately feedback linearizable systems we introduce a perturbation parameter by applying high gain feedback and use both the feedback linearization method and the singular perturbation method for the controller design through this we can overcome the rigorous conditions of the feedback linearization method and can reduce the dimension of the slow model of the singularly perturbed system

slicing skinning and grafting we prove that a bers slice is never algebraic meaning that its zariski closure in the character variety has strictly larger dimension a corollary is that skinning maps are never constant the proof uses grafting and the theory of complex projective structures

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

on character varieties sets of discrete characters and nonzero degree maps a it knot manifold is a compact connected irreducible orientable 3 manifold whose boundary is an incompressible torus we first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps these results are applied later in the paper where we fix a small knot manifold m and investigate various sets of characters of representations rho pi 1 m to rm psl 2 bbb c whose images are discrete we show that the topology of these sets is intimately related to the algebraic structure of the rm psl 2 bbb c character variety of m as well as dominations of manifolds by m and its dehn fillings we apply our results to the following question of shicheng wang it are nonzero degree maps between infinitely many distinct dehn fillings of two hyperbolic knot manifolds m and n induced by a nonzero degree map m to n we show that the answer is yes generically using this we show that if a small mathcal h minimal hyperbolic knot manifold admits non homeomorphic mathcal h minimal dehn fillings it admits infinitely many such fillings we also construct the first infinite families of small closed connected orientable manifolds which are minimal in the sense that they do not admit nonzero degree maps other than homotopy equivalences to any aspherical manifold

repetitive control of hamiltonian systems based on variational symmetry this paper is concerned with repetitive control of hamiltonian systems which is based on iterative learning control utilizing the variational symmetry of those systems variational symmetry allows us to obtain an algorithm to solve a certain class of optimal control problems in a repetitive control framework therefore the proposed method can deal with not only trajectory tracking control problems but also optimal trajectory generation problems never before considered in a repetitive control framework a convergence analysis of this algorithm is also discussed furthermore some numerical simulations demonstrate the effectiveness of the proposed method

amenability poincare series and quasiconformal maps any coveringy x of a hyperbolic riemann surfacex of finite area determines an inclusion of teichmuller spaces teich x teich y we show this map is an isometry for the teichmuller metric if the covering isamenable and contracting otherwise in particular we establish 1 for classical poincare series kra s theta conjecture

the monodromy groups of schwarzian equations on closed riemann surfaces let 1 r psl 2 be a homomorphism of the fundamental group of an oriented closed surface r of genus exceeding one we will establish the following theorem theorem necessary and sufficient for to be the monodromy representation associated with a complex projective stucture on r either unbranched or with a single branch point of order 2 is that 1 r be nonelementary a branch point is required if and only if the representation does not lift to sl 2

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

nonlinear repetitive control repetitive controllers are generally applied to reject periodic disturbances and to track periodic reference signals with a known period their design is based on the internal model principle proposed by francis and wonham 1975 this paper describes a new finite dimensional siso repetitive controller for two different classes of nonlinear plants simulation results show asymptotic tracking of the periodic reference signal by the proposed repetitive controller in closed loop up to the nth harmonic frequency a proof of robustness of the repetitive control system to small nonlinearities like actuator nonlinearities is provided

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

skinning maps are finite to one we show that thurston s skinning maps of teichmuller space have finite fibers the proof centers around a study of two subvarieties of the rm sl 2 mathbb c character variety of a surface one associated with complex projective structures and the other associated with a 3 manifold using the morgan shalen compactification of the character variety and author s results on holonomy limits of complex projective structures we show that these subvarieties have only a discrete set of intersections

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

nonlinear control via approximate input output linearization the ball and beam example approximate input output linearization of nonlinear systems which fail to have a well defined relative degree is studied for such systems a method for constructing approximate systems that are input output linearizable is provided the analysis presented is motivated through its application to a common undergraduate control laboratory experiment the ball and beam where it is shown to be more effective for trajectory tracking than the standard jacobian linearization

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

word hyperbolic dehn surgery in the late 1970 s thurston dramatically changed the nature of 3 manifold theory with the introduction of his geometrisation conjecture and by proving it in the case of haken 3 manifolds 23 the conjecture for general closed orientable 3 manifolds remains perhaps the most important unsolved problem in the subject a weaker form of the conjecture 19 deals with the fundamental group of a closed orientable 3 manifold it proposes that either it contains z z as a subgroup or it is word hyperbolic in the sense of gromov 11 word hyperbolic groups are precisely those groups which satisfy a linear isoperimetric inequality they are of fundamental importance in geometric group theory and have very many useful properties

convergence of freely decomposable kleinian groups we consider a compact orientable hyperbolic 3 manifold with a compressible boundary suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination we prove that if this limit projective lamination is doubly incompressible then the sequence has compact closure in the deformation space as a consequence we generalise thurston s double limit theorem and solve his conjecture on convergence of function groups affirmatively

a sequence of finite dimensional controllers for repetitive control of positive real systems in this paper we analyze the performance of positive real plants coupled with a progressively increasing number of harmonic oscillators we prove that as the number of oscillators tends to infinity thus covering the all frequency range the steady state output weakly converges to the periodic signal to be tracked on the basis of this result we next show that under a suitable constraint on the signal to be tracked strong convergence is actually achieved

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

convexes divisibles iv structure du bord en dimension 3 divisible convex sets iv boundary structure in dimension 3 let be an indecomposable properly convex open subset of the real projective 3 space which is divisible i e for which there exists a torsion free discrete group of projective transformations preserving such that the quotient m is compact we study the structure of m and of when is not strictly convex the union of the properly embedded triangles in projects in m onto an union of finitely many disjoint tori and klein bottles which induces an atoroidal decomposition of m every non extremal point of is on an edge of a unique properly embedded triangle in and the set of vertices of these triangles is dense in the boundary of see figs 1 to 4 moreover we construct examples of such divisible convex open sets

the classification of kleinian surface groups i models and bounds we give the first part of a proof of thurston s ending lamination conjecture in this part we show how to construct from the end invariants of a kleinian surface group a lipschitz model for the thick part of the corresponding hyperbolic manifold this enables us to describe the topological structure of the thick part and to give a priori geometric bounds

uniqueness of limit cycles for lienard differential equations of degree four abstract we prove that any classical lienard differential equation of degree four has at most one limit cycle and the limit cycle is hyperbolic if it exists this result gives a positive answer to the conjecture by a lins w de melo and c c pugh 1977 4 about the number of limit cycles for polynomial lienard differential equations for n 4

a symplectic map between hyperbolic and complex teichmuller theory let s be a closed orientable surface of genus at least 2 the space th ml where th is the hyperbolic teichmuller space of s and ml is the space of measured geodesic laminations on s is naturally a real symplectic manifold the space cp of complex projective structures on s is a complex symplectic manifold a relation between these spaces is provided by thurston s grafting map gr we prove that this map although not smooth is symplectic the proof uses a variant of the renormalized volume defined for hyperbolic ends

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

brief robust learning control for a class of nonlinear systems with periodic and aperiodic uncertainties this paper addresses the robust learning control problem for a class of nonlinear systems with structured periodic and unstructured aperiodic uncertainties a recursive technique is proposed which extends the backstepping idea to the robust repetitive learning control systems a learning evaluation function instead of a lyapunov function is formulated as a guideline for derivation of the control strategy which guarantees the asymptotic stability of the tracking system a design example is given

the classification of kleinian surface groups ii the ending lamination conjecture thurston s ending lamination conjecture states that a hyperbolic 3manifold n with nitely generated fundamental group is uniquely determined by its topological type and its end invariants in this paper we prove this conjecture for kleinian surface groups the general case when n has incompressible ends relative to its cusps follows readily the main ingredient is a uniformly bilipschitz model for the quotient of h 3 by a kleinian surface group

the weil petersson symplectic structure at thurston s boundary the weil petersson kahler structure on the teichmuller space f of a punctured surface is shown to extend in an appropriate sense to thurston s symplectic structure on the space m f 0 of measured foliations of compact support on the surface we introduce a space m f 0 of decorated measured foliations whose relationship to m f 0 is analogous to the relationship between the decorated teichmuller space f and f m f 0 is parametrized by a vector space and there is a naturl piecewise linear embedding of m f 0 in m f 0 which pulls back a global differential form to thurston s symplectic form we exhibit a homeomorphism between f and m f 0 which preseres the natural two forms on these spaces following thurston we finally consider the space y of all suitable classes of metrics of constant gaussian curvature on the surface form a natural completion y of y and identify y y with m f 0

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

measured foliations and the minimal norm property for quadratic differentials the objective is to prove two important inequalities of riemann surface theory and to present them in such a way that one can see their close relationship the first inequality is the minimal norm property of quadratic differentials first observed by marden and strebel 8 actually a somewhat different inequality is given the minimum norm properties have many applications one of which is a proof of the result of hubbard and masur 7 concerning the existence of a unique holomorphic quadratic differential whose horizontal foliation and vertical measure realize a given measure class of measured foliations on a given riemann surface this result is given in section 7 the method of proof is quite different from the one in 7 the second important inequality is the main inequality of reich and strebel it has central importance in teichmiiller theory and can be used to derive the infinitesimal form of teichmiiller s metric 10 moreover this inequality can be used to show that teichmiiller s metric is the integral of its infinitesimal form 6 a proof of this result by other methods was given by o byrne in 9 both of these inequalities turn out to be consequences of certain properties of the trajectory structure of a holomorphic quadratic differential for the purposes of the main inequality we are able to bypass some of the theory of trajectories by means of an averaging device used by teichmiiller this averaging device also appears in the proofs of teichmiiller s theorem given by bers 4 and abikoff 1 but for the minimum norm property in the form given by marden and strebel 8 the detailed theory of trajectories seems to be necessary

lattice point asymptotics and volume growth on teichmuller space we apply some of the ideas of the ph d thesis of g a margulis mar70 to teichmuller space let x be a point in teichmuller space and let br x be the ball of radius r centered at x with distances measured in the teichmuller metric we obtain asymptotic formulas as r tends to infinity for the volume of br x and also for the cardinality of the intersection of br x with an orbit of the mapping class group

a proof of the finite filling conjecture let m be a compact connected orientable hyperbolic 3 manifold whose boundary is a torus we show that there are at most five slopes on m whose associated dehn fillings have either a finite or an infinite cyclic fundamental group furthermore the distance between two slopes yielding such manifolds is no more than three and there is at most one pair of slopes which realize the distance three each of these bounds is realized when m is taken to be the exterior of the figure 8 sister knot

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