geometric group actions on trees we define geometric group actions on r trees as dual to a measured foliation on a 2 complex with some finiteness and injectivity properties we prove that an action is nongeometric if and only if it is a nontrivial strong limit in the sense of gillet shalen we give a simple new construction of the bass serre tree of a graph of groups and we show that a simplicial action is geometric if and only if edge groups are finitely generated we prove that geometric actions with trivial edge stabilizers have finitely many orbits of branch points and finite rank introduction an r tree is an arcwise connected metric space in which every arc is isometric to an interval of r see for instance shal sha2 mor for historical remarks references and motivation it is well known that codimension one measured foliations or laminations on compact manifolds have strong connections with group actions on r trees

free vibration analysis of a uniform multi span beam carrying multiple spring mass systems abstract the literature regarding the free vibration analysis of single span beams carrying a number of spring mass systems is plenty but that of multi span beams carrying multiple spring mass systems is fewer thus this paper aims at determining the exact solutions for the natural frequencies and mode shapes of a uniform multi span beam carrying multiple spring mass systems firstly the coefficient matrices for an intermediate pinned support an intermediate spring mass system left end support and right end support of a uniform beam are derived next the numerical assembly technique for the conventional finite element method is used to establish the overall coefficient matrix for the whole vibrating system finally equating the last overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes in this paper the natural frequencies and associated mode shapes of the vibrating system are obtained directly from the differential equation of motion of the continuous beam and no other assumptions are made thus the last solutions are the exact ones the effects of attached spring mass systems on the free vibration characteristics of the 1 4 span beams are studied

positive configurations of flags in a building and limits of positive representations parreau compactified the hitchin component of a closed surface s of negative euler characteristic in such a way that a boundary point corresponds to the projectivized length spectrum of an action of pi 1 s on an mathbb r euclidean building in this paper we use the positivity properties of hitchin representations introduced by fock and goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

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

determining thurston classes using nielsen types in previous work 3 we showed how the thurston or bers classifications of diffeomorphisms of surfaces could be obtained using the nielsen types of the lifts of the diffeomorphism to the unit disc in this paper we find improved conditions on the nielsen types for the thurston and bers classes we use them to verify that an example studied by nielsen is a pseudo anosov diffeomorphism with stretching factor of degree 4 this example is of interest in its own right but it also serves to illustrate exactly how the nielsen types are used for verifying examples we discuss the general usefulness of this method

dehn twists and pseudo anosov diffeomorphisms we show that it is possible to obtain many pseudo anosov diffeomorphisms from dehn twists in particular we generalize a theorem of long and morton to obtain that iff is a pseudo anosov diffeomorphism of an oriented surface andt is the dehn twist around the simple closed curve then the isotopy class oft n f contains a pseudo anosov diffeomorphism except for at most 7 consecutive values ofn

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

sur l unique ergodicite des 1 formes fermees singulieres we study the relations between the de rham cohomology class of a closed differential 1 form with morse singularities on a manifoldm of dimensionn 3 and the ergodic properties of the foliationf it defines we show by examples that if the fundamental group is large enough very different behaviours can occur in the same class in contrast if the fundamental group admits no surjective homomorphism onto the free group on 3 generators thenf is always uniquely ergodic provided it has no compact leaf if the natural homomorphism 1 m h1 m z torsion does not factor through a free group then the same result is true in almost every cohomology class we also give results about the existence of noncompact leaves and the number of ergodic measures

fixed points of pseudo anosov diffeomorphisms of surfaces this article supplements richard miller s paper 6 by establishing a result on the minimum number of fixed points in the homotopy class of a pseudo anosov surface transformation we thus verify one of the many farreaching results that william thurston announced in 1977 in a preprint s dealing with the classification up to homotopy of diffeomorphisms of compact surfaces thurston stated his theorems without proofs but with many hints at techniques fathi laudenbach poenaru and other workers at an intensive seminar at the university of orsay gave detailed proofs of some but not all of thurston s assertions and published their findings in a monograph 7 more recently miller 6 and gilman 2 have verified thurston s claim that his theorem 4 s p 81 follows from the classical investigations of jakob nielsen 4 using miller s exposition of the nielsenthurston theory we establish

mixed zero sum stochastic differential game and american game options in this paper we solve the mixed zero sum stochastic differential game problem in the general case the main tool is the notion of a local solution of backward stochastic differential equations bsdes with two reflecting barriers as an application we deal with the american game options

on the nielsen type and the classification for the mapping class group let s be a compact surface of genus g with n boundary components with 2g 2 n 0 and let m s be its mapping class group also known as the teichmiiller modular group using measurd foliations thurston 15 has shown that m s contains three different types of elements bers has shown this decomposition can be obtained using teichmiiller distance 3 subdividing one of the classes further he obtains four classes nielsen has a series of papers 7 8 parts i iii in which he develops an elaborate theory of fixed points of mapping classes if t is any homeomorphism of s and h a lift of t or of a power of 1 to the upper half plane nielsen s theory involves the assignment of a pair of integers to h called the nielsen type of h the purpose of this paper is to show how the thurston classes and the bers classes can be defined in terms of the nielsen types of the lifts of

numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the rayleigh ritz method unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction for plate structures this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core this paper investigates the use of simple and orthogonal polynomials in the rayleigh ritz method for unilateral plate buckling the orthogonal polynomials used are the classical chebyshev types 1 and 2 legrende hermite and laguerre the study presents a comparison between the efficiency of the polynomial based displacement functions with regard to elastic bilateral and unilateral plate buckling where efficiency is measured as a function of their convergence characteristics some buckling solutions for plates with varying boundary conditions and in plane shear loads are also provided as an illustration copyright 1999 john wiley sons ltd

la dynamique des pseudogroupes de rotations we study dynamical systems on the circle generated by a finite number of partially defined rotations we construct new examples with all orbits dense this leads to non simplicial free actions of free groups on trees we study the generic dynamics for these pseudogroups and their 1 parameter families we show that in suitable 2 parameter families the set of pseudogroups having a dense orbit is a sierpinski curve we generalize results on interval exchange transformations obtained by boshernitzan veech rips

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

two player nonzero sum stopping games in discrete time we prove that every two player non zero sum stopping game in discrete time admits an equilibrium in randomized strategies for every 0 we use a stochastic variation of ramsey theorem which enables us to reduce the problem to that of studying properties of equilibria in a simple class of stochastic games with finite state space

on equalizers of sections abstract let g and m be groups and a b g g m group theoretic sections of the natural projection g m g we use the almost stability theorem pro trees and new folding sequence techniques to show that if g is finitely generated then the equalizer of a and b is a free factor of g which answers a question of g m bergman

on the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements abstract this paper is concerned with a numerical convergence study of three discretization methods rayleigh ritz galerkin and finite element as applied to the analysis of free bending linear vibrations of a uniform bernoulli euler beam carrying inertia elements at intermediate points a cantilever beam carrying a lumped mass with rotary inertia at an arbitrary intermediate point and another at the beam tip is used as a case study the first five frequency parameters of this beam obtained by using the rayleigh ritz method in conjunction with broadly admissible base beam eigenfunctions galerkin s method in conjunction with strictly admissible base beam eigenfunctions and the classical finite element method are compared with the exact ones over a wide range of values of the system parameters conclusions are drawn regarding the effects of the various system parameters on the accuracy and convergence characteristics of the above three discrete methods and on the mode shapes and the first five frequency parameters of the beam system

universal spaces for r trees r trees arise naturally in the study of groups of isometries of hyperboiic space an r tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals it follows that an r tree is locally arcwise connected contractible and one dimensional unique and local arcwise connectivity characterize r trees among metric spaces a universal r tree would be of interest in attempting to classify the actions of groups of isometries on r trees it is easy to see that there is no universal r tree however we show that there is a universal separable r tree t 0

deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod ulus 1 the functions we consider are the corresponding eigen functions in theorem 1 1 we prove that the limit inferior of the ergodic sums n x 0 x n 1 n n is bounded for every point x in the phase space in theorem 1 2 we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length under additional assumptions we prove that ergodic integrals satisfy the central limit theorem theorem 1 3 theorem 1 9

covering r trees r free groups and dendrites abstract we prove that every length space x is the orbit space with the quotient metric of an r tree x via a free action of a locally free subgroup x of isometries of x the mapping x x is a kind of generalized covering map called a url map and is universal among url maps onto x x is the unique r tree admitting a url map onto x when x is a complete riemannian manifold m n of dimension n 2 the menger sponge the sierpin ski carpet or gasket x is isometric to the so called universal r tree a c which has valency c 2 0 at each point in these cases and when x is the hawaiian earring h the action of x on x gives examples in addition to those of dunwoody and zastrow that negatively answer a question of j w morgan about group actions on r trees indeed for one length metric on h we obtain precisely zastrow s example

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

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

a topological characterization of r trees r trees arise naturally in the study of groups of isometries of hyperbolic space an r tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals r actions on r trees can be viewed as ideal points in the compactification of groups of isometries as such they have applications to the study of hyperbolic manifolds our concern in this paper however is with the topological characterization of r trees our main theorem is the following let x p be a metric space then x is uniquely arcwise connected and locally arcwise connected if and only ijf x admits a compatible metric d such that x d is an r tree essentially we show how to put a convex metric on a uniquely arcwise connected locally arcwise connected metrizable space

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

hausdorff dimension bounds for smoothness of holonomies for codimension 1 hyperbolic dynamics we prove that the stable holonomies of a proper codimension 1 attractor lambda for a c r diffeomorphism f of a surface are not c1 theta for theta greater than the hausdorff dimension of the stable leaves of f intersected with lambda to prove this result we show that there are no diffeomorphisms of surfaces with a proper codimension 1 attractor that are affine on a neighbourhood of the attractor and have affine stable holonomies on the attractor

the controller and stopper game for a linear diffusion consider a process x x t 0 t 1 with values in the interval i 0 1 absorption at the boundary points of i and dynamics dx t t dt t dw t x 0 x the values t t are selected by a controller from a subset of 0 1 that depends on the current position x t for every t 0 at any stopping rule of his choice a second player called stopper can halt the evolution of the process x upon which he receives from the controller the amount e u x here 2 0 1 is a discount factor and u 0 1 is a continuous reward function under appropriate conditions on this function and on the contoller s set of choices it is shown that the two players have a saddle point of optimal strategies these can be described fairly explicitly by reduction to a suitable problem of optimal stopping whose maximal expected reward v coincides with the value of the game v sup inf x e e u x inf x sup e e u x

free actions on ℝ trees we characterize the free minimal actions of finitely generated groups on r trees in terms of certain equivalence relations on compact metric graphs

dual garside structure and reducibility of braids abstract benardete gutierrez and nitecki showed an important result which relates the geometrical properties of a braid as a homeomorphism of the punctured disk to its algebraic garside theoretical properties namely they showed that if a braid sends a standard curve to another standard curve then the image of this curve under the action of each factor of the left normal form of the braid with the classical garside structure is also standard we provide a new simple geometric proof of the result by benardete gutierrez nitecki which can be easily adapted to the case of the dual garside structure of braid groups with the appropriate definition of standard curves in the dual setting this yields a new algorithm for determining the nielsen thurston type of braids

1 formes fermees singulieres et groupe fondamental we study the influence of the 1 of a closed manifoldm n n 3 on the foliations ofm defined by closed differential 1 forms with morse singularities of index 0 n every nonexact form is cohomologous to a weakly complete one that is one whose leaf space is of the same type as that of a nonsingular form generically a form has compact leaves or is weakly complete if 1 m has no quotient isomorphic to then every nonexact form onm is weakly complete we also say a form is complete if every path inm is homotopic to either a path transverse to or a path contained in a leaf of completeness of depends only on its de r ham cohomology class the set of complete cohomology classes depends only on 1 m and is related to finitely generated normal subgroups of 1 m with quotient if 1 m is nilpotent or even polycyclic every nonexact form onm is complete on irreducible 3 manifolds a form is complete iff it is cohomologous to a nonsingular one

exact and approximate nash equilibria in discounted markov stopping games with terminal redemption abstract this work is concerned with a class of discrete time zero sum games with two players and markov transitions on a denumerable space the general structure of the game is as follows before the game starts player i receives a bond from player ii and at each decision time player ii can stop the system paying a terminal bond to player i or if the game is not halted player i selects an action to drive the system and receives a running reward bond from player ii at termination player i redeems the accumulated bonds and the performance of a pair of decision strategies is measured by the expectation of the value at time zero of the bonds accumulated by player i under standard continuity compactness conditions the following conclusions are established i it is shown that this stopping game has a value function which is characterized by an equilibrium equation and ii such a result is used to obtain the existence of a nash equilibrium also iii the method of successive approximations is used to construct approximate nash equilibria for the game

a recipe for short word pseudo anosovs given any generating set of any subgroup g of the mapping class group of a surface we find an element f with word length bounded by a constant k depending only on the surface and with the property that the minimal subsurface supporting a power of f is as large as possible for elements of g in particular if g contains a pseudo anosov map we find one of word length at most k we also find new examples of convex cocompact free subgroups of the mapping class group

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