holomorphic differentials thermostats and anosov flows we introduce a new family of thermostat flows on the unit tangent bundle of an oriented riemannian two manifold suitably reparametrised these flows include the geodesic flow of metrics of negative gauss curvature and the geodesic flow induced by the hilbert metric on the quotient surface of divisible convex sets we show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties in particular we prove that they admit a dominated splitting and we identify special cases in which the flows are anosov in the latter case we study when they admit an invariant measure in the lebesgue class and the regularity of the weak foliations

the margulis lemma and the thick and thin decomposition for convex real projective surfaces convex real projective surfaces are quotients of simply convex domains in the real projective planerp2under the properly discontinuous and free action of a projective automorphism group such surfaces carry hilbert metrics defined by logarithms of cross ratios we prove an analogous proposition to the margulis lemma in hyperbolic geometry holding for such a surface with a hilbert metric this allows us to decompose into thick and thin components we show a compactness result that given a certain collection of simple closed curves 1 non the subset of the deformation space of convex structures p corresponding to structures where the hilbert lengths of 1 nare bounded above is compact

runge kutta solutions of a hyperbolic conservation law with source term spurious long term solutions of a finite difference method for a hyperbolic conservation law with a general nonlinear source term are studied results are contrasted with those that have been established for nonlinear ordinary differential equations various types of spurious behavior are examined including spatially uniform equilibria that exist for arbitrarily small time steps nonsmooth steady states with profiles that jump between fixed levels and solutions with oscillations that arise from nonnormality and exist only in finite precision arithmetic it appears that spurious behavior is associated in general with insufficient spatial resolution the potential for curbing spuriosity by using adaptivity in space or time is also considered

computational chaos a prelude to computational instability abstract chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment in two simple examples we find that as increases chaos first sets in when attractor a acquires two distinct points that map to the same point this happens when a acquires slopes of the same sign in a rectifying coordinate system at two consecutive intersections with the critical curve chaotic and quasi periodic behavior may then alternate within a range of before computational instability finally prevails bifurcations to and from chaos and transitions to computational instability are highly scheme dependent even among differencing schemes of the same order systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

equilibria of runge kutta methods it is known that certain runge kutta methods share the property that in a constant step implementation if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system such methods are calledregular in the present paper we provide a recursive test to check whether given method is regular moreover by examining solution trajectories of linear equations we prove that the order of ans stage regular method may not exceed 2 s 2 2 and that the maximal order of regular runge kutta method with an irreducible stability function is 4

regularity properties of one leg methods for delay differential equations abstract the main purpose of this paper is to investigate the asymptotic states of one leg methods for multidelay differential equations in particular the existence of spurious steady solutions and period 2 solutions in constant or variable timestep is studied and the concepts of r 1 regularity and r 2 regularity of one leg methods for multidelay differential equations are introduced and studied some conditions guaranteeing r 1 regularity and r 2 regularity of such methods applied to multidelay differential equations with some important structures are given

on convex projective manifolds and cusps this study of properly or strictly convex real projective manifolds introduces notions of parabolic horosphere and cusp results include a margulis lemma and in the strictly convex case a thick thin decomposition finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds this is proved using a characterisation of ellipsoids in projective space except in dimension 3 there are only finitely many topological types of strictly convex man ifolds with bounded volume in dimension 4 and higher the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part there is an algebraic characterization of strict convexity in terms of relative hyperbolicity surfaces are ubiquitous throughout mathematics in good measure because of the geometry of riemann surfaces similarly thurston s insights into the geometry of 3 manifolds have led to many developments in diverse areas this paper develops the bridge between real projective geometry and low dimensional topology real projective geometry is a rich subject with many connections in recent years it has been combined with topology in the study of projective structures on manifolds classically it provides a unifying framework as it contains the three constant curvature geometries as subgeometries in dimension 3 it contains the eight thurston geometries up to a subgroup of index 2 in the case of product geometries and there are paths of projective structures that correspond to transitions between different thurston geometries on a fixed manifold moreover there is a link between real projective deformations and complex hyperbolic deformations of a real hyperbolic orbifold see 22 projective geometry therefore offers a general and versatile viewpoint for the study of 3 manifolds another window to projective geometry the symmetric space sl n r so n is isomorphic to the group of projective automorphisms of the convex set in projective space obtained from the open cone of positive definite quadratic forms in n variables this set is properly convex its closure is a compact convex set which is disjoint from some projective hyperplane the boundary of the closure has a rich structure as it consists of semi definite forms and when n 3 contains a dense set of flat 2 discs each corresponding to a family of semi definite forms of rank 2 which may be identified with a copy of the hyperbolic plane from a geometrical point of view there is a crucial distinction between strictly convex domains which contain no straight line segment in the boundary and the more general class of properly convex domains the former behave like manifolds of negative sectional curvature and the latter like arbitrary symmetric spaces however projective manifolds are more general kapovich 34 has shown that there are closed strictly convex 4 manifolds which do not admit a hyperbolic structure the hilbert metric is a complete finsler metric on a properly convex set this is the hyperbolic metric in the klein model when is a round ball a simplex with the hilbert metric is isometric to a normed vector space and appears in a natural geometry on the lie algebra sln a singular version of this metric arises in the study of certain limits of projective structures the hilbert metric has a hausdorff measure and hence a notion of finite volume

topological tameness of margulis spacetimes we show that margulis spacetimes without parabolic holonomy elements are topologically tame a margulis spacetime is the quotient of the 3 dimensional minkowski space by a free proper isometric action of the free group of rank geq 2 we will use our particular point of view that the margulis spacetime is a manifold with boundary with an bbb r sf p 3 structure in an essential way the basic tools are a bordification by a closed bbb r sf p 2 surface with free holonomy group and the work of goldman labourie and margulis on geodesics in the margulis spacetimes and 3 manifold topology

the deformation spaces of convex rp structures on 2 orbifolds we determine that the deformation space of convex real projective structures that is projectively flat torsion free connections with the geodesic convexity property on a compact 2 orbifold of negative euler characteristic is homeomorphic to a cell of certain dimension the basic techniques are from thurston s lecture notes on hyperbolic 2 orbifolds the previous work of goldman on convex real projective structures on surfaces and some classical geometry

does error control suppress spuriosity in the numerical solution of initial value ordinary differential equations to what extent does local error control confer global properties this work concentrates on global steady states or fixed points it is shown that for systems of equations spurious fixed points generally cease to exist when local error control is used for scalar problems on the other hand locally adaptive algorithms generally avoid spurious fixed points by an indirect method the stepsize selection process causes spurious fixed points to be unstable however problem classes exist where for arbitrarily small tolerances stable spurious fixed points persist with significant basins of attraction a technique is derived for generating such examples

spurious behavior of a symplectic integrator in this article we study the existence and behavior of spurious solutions of symplectic euler method for some hamiltonian systems it is shown that the symplectic integrator applied to hamiltonian systems in general doesn t avoid spurious behavior even spurious period two solutions the numerical results are presented

deformation spaces associated to compact hyperbolic manifolds in this paper we take a first step toward understanding representations of cocompact lattices in so n 1 into arbitrary lie groups by studying the deformations of rational representations see proposition 5 1 for a rather general existence result this proposition has a number of algebraic applications for example we remark that such deformations show that the margulis super rigidity theorem see 30 cannot be extended to the rank 1 case we remark also that if so n 1 is one of the standard arithmetic examples described in section 7 then has a faithful representation in so n 1 the galois conjugate of the uniformization representation and proposition 5 1 may be used to deform the direct sum of and the trivial representation in so n 2 thereby constructing non trivial families of irreducible orthogonal representations of however most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain

preservation of hopf bifurcation for neutral delay differential equations by θ methods this paper considers the preservation of hopf bifurcation of some neural delay differential equations by q method by analyzing the dynamics of the numerical discrete system derived by q method we show that q method could inherit the hopf bifurcation and the asymptotical stability for sufficiently small stepsize h 1 m where m is a positive integer in particular for q 1 2 the result holds for any stepsize h 1 m furthermore the stability of the closed invariant curve is established finally some numerical examples are illustrated to support the analytic results

the degeneration of convex ℝℙ2 structures on surfaces let m be a compact surface of negative euler characteristic and let c m be the deformation space of convex real projective structures on m with positive hyperbolic holonomy for every choice of pants decomposition for m there is a well known parameterization of c m known as the goldman parameterization in this paper we study how some geometric properties of the real projective structure on m degenerate as we deform it so that the internal parameters of the goldman parameterization leave every compact set while the boundary invariants remain bounded away from zero and infinity

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

dynamical approach study of spurious steady state numerical solutions of nonlinear differential equations i the dynamics of time discretization and its implications for algorithm development in computational fluid dynamics abstract the goal of this paper is to utilize the theory of nonlinear dynamics approach to investigate the possible sources of errors and slow convergence and nonconvergence of steady state numerical solutions when using the time dependent approach for nonlinear hyperbolic and parabolic partial differential equations terms this interdisciplinary research belongs to a subset of a new field of study in numerical analysis sometimes referred to as the dynamics of numerics and the numerics of dynamics at the present time this new interdisciplinary topic is still the property of an isolated discipline with all too little effort spent in pointing out an underlying generality that could make it adaptable to diverse fields of applications this is the first of a series of research papers under the same topic our hope is to reach researchers in the fields of computational fluid dynamics cfd and in particular hypersonic and combustion related cfd by simple examples in which the exact solutions of the governing equations are known the application of the apparently straightforward numerical technique to genuinely nonlinear problems can be shown to lead to incorrect or misleading results one striking phenomenon is that with the same initial data the continuum and its discretized counterpart can asymptotically approach different stable solutions this behavior is especially important for employing a time dependent approach to the steady state since the initial data are usually not known and a freestream condition or an intelligent guess for the initial conditions is often used with the unique property of the different dependence of the solution on initial data for the partial differential equation and the discretized counterpart it is not easy to delineate the true physics from numerical artifacts when numerical methods are the sole source of solution procedure for the continuum part i concentrates on the dynamical behavior of time discretization for scalar nonlinear ordinary differential equations in order to motivate this new yet unconventional approach to algorithm development in cfd and to serve as an introduction for parts 11 and iii of the same series of research papers

surface group representations with maximal toledo invariant we develop the theory of maximal representations of the fundamental group 1 of a compact connected oriented surface possibly with boundary into lie groups g of hermitian type for any homomorphism p 1 g we define the toledo invariant t p a numerical invariant which has both topological and analytical interpretations we establish important properties of t among which continuity uniform boundedness on the representation variety additivity under connected sum of surfaces and congruence relations mod we thus obtain information about the representation variety as well as striking geometric properties of maximal representations that is representations whose toledo invariant achieves the maximum value moreover we establish properties of boundary maps associated to maximal representations which generalize naturally monotonicity properties of semiconjugations of the circle we define a rotation number function for general locally compact groups and study it in detail for groups of hermitian type properties of the rotation number together with the existence of boundary maps lead to additional invariants for maximal representations and show that the subset of maximal representations is always real semialgebraic

exact wkb and abelianization for the t_3 equation we describe the exact wkb method from the point of view of abelianization both for schrodinger operators and for their higher order analogues opers the main new example which we consider is the t 3 equation an order 3 equation on the thrice punctured sphere with regular singularities at the punctures in this case the exact wkb analysis leads to consideration of a new sort of darboux coordinate system on a moduli space of flat mathrm sl 3 connections we give the simplest example of such a coordinate system and verify numerically that in these coordinates the monodromy of the t 3 equation has the expected asymptotic properties we also briefly revisit the schrodinger equation with cubic potential and the mathieu equation from the point of view of abelianization

difference schemes of optimum degree of implicitness for a family of simple odes with blow up solutions abstract it has been speculated that the linearly implicit linearized trapezoidal rule should be especially useful for computing blow up solutions since it features discrete blow up we study the question of how implicit a scheme should be from several different points of view we discuss further the properties of the linearized trapezoidal and of its discrete blow up we give exact schemes for a family of odes with polynomial nonlinearity and that shows the optimum degree of implicitness they need we show that other standard schemes the trapezoidal rule itself and the implicit midpoint rule are exact on certain differential equations we compare for several schemes and different nonlinearities the size of the leading error terms and we briefly discuss two ways of adapting the degree of implicitness of a scheme to a given differential equation

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

chaos free numerical solutions of reaction diffusion equations two numerical methods which do not bring contrived chaos into the solution are proposed for the solution of the riccati logistic equation though implicit in nature with the resulting improvements in stability the methods are applied explicitly when extended to the numerical solution of fisher s equation in which the quadratic polynomial representing the derivative in the riccati equation appears as the reaction term the solution is found by solving a linear system of algebraic equations at each time step as opposed to solving a nonlinear system which frequently happens when solving nonlinear partial differential equations the approaches adopted are extended to an ordinary differential equation in which the derivative is expressed as a cubic polynomial in the dependent variable the solution of this initial value problem is not available in closed form for finite values of the independent variable t under the conditions stated numerical solutions are seen to converge to the correct steady state solution a nonlinear partial differential equation which governs the conduction of electrical impulses along a nerve axon and which has the aforementioned cubic polynomial as its reaction term is solved by applying the numerical methods developed for solving the ordinary differential equation the solution to this nonlinear reaction diffusion equation is determined by solving a linear algebraic system at each time step

poisson geometry of sl 3 c character varieties relative to a surface with boundary the sl 3 c representation variety r of a free group f r arises naturally by considering surface group representations for a surface with boundary there is an sl 3 c action on the coordinate ring of r by conjugation the geometric points of the subring of invariants of this action is an affine variety x the points of x parametrize isomorphism classes of completely reducible representations we show the coordinate ring c x is a complex poisson algebra with respect to a presentation of f r imposed by the surface lastly we work out the bracket on all generators when the surface is a three holed sphere or a one holed torus

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 unified approach to spurious solutions introduced by time discretization part i basic theory the asymptotic states of numerical methods for initial value problems are examined in particular spurious steady solutions solutions with period 2 in the timestep and spurious invariant curves

euler like discrete models of the logistic differential equation abstract to understand the behavior of difference schemes on nonlinear differential equations it seems desirable to extend the standard linear stability theory into a nonlinear theory as a step in that direction we investigate the stability properties of euler related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other we find that these schemes are superior to explicit schemes when they are stable and the blow up time has not passed for these h values they are dynamically faithful when these schemes turn unstable however they have much less desirable properties than explicit or fully implicit schemes they become simultaneously superstable and unstable this is explained by the fact that these schemes are not self adjoint the linearly implicit self adjoint scheme is dynamically faithful in an euler typical range of step sizes and gives correct stability for all step sizes

piecewise linearized and linearized θ methods for ordinary and partial differential equations lnitial and boundary value problems appear frequently in many branches of physics in this paper several numerical methods based on linearization techniques for solving these prob lems are reviewed first piecewise linearized methods and linearized 0 methods are considered for the solution of initial value problems in ordinary differential equations second piecewise linearized techniques for two point boundary value problems in ordinary differential equations are developed and used in conjunction with a shooting method in order to overcome the lack of convergence asso ciated with shooting piecewise linearized methods which provide piecewise analytical solutions and yield nonstandard finite difference schemes axe presented third methods of lines in either space or time for the solution of one dimensional convection reaction diffusion problems that transform the original problem into an initial or boundary value one are reviewed methods of lines in time that result in boundary value problems at each time step can be solved by means of the techniques described here whereas methods of lines in space that yield initial value problems and employ ei ther piecewise linearized techniques or linearized 0 methods in time are also developed finally for multidimensional problems approximate factorization methods are first used to transform the multi dimensional problem into a sequence of one dimensional ones which are then solved by means of the linearized and piecewise linearized methods presented here 2003 elsevier science ltd all rights reserved

minimal lagrangian connections on compact surfaces abstract we introduce the notion of a minimal lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non vanishing euler characteristic combining our classification with results of labourie and loftin we conclude that every properly convex projective surface arises from a unique minimal lagrangian connection

anosov representations domains of discontinuity and applications the notion of anosov representations has been introduced by labourie in his study of the hitchin component for sl n r subsequently anosov representations have been studied mainly for surface groups in particular in the context of higher teichmuller spaces and for lattices in so 1 n in this article we extend the notion of anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties in particular given an anosov representation g we explicitly construct open subsets of compact g spaces on which acts properly discontinuously and with compact quotient as a consequence we show that higher teichmuller spaces parametrize locally homogeneous geometric structures on compact manifolds we also obtain applications regarding non standard compact clifford klein forms and compactifications of locally symmetric spaces of infinite volume

nonlinear instability in dissipative finite difference schemes a unified analysis of reaction diffusion equations and their finite difference representations is presented the parallel treatment of the two problems shows clearly when and why the finite difference approximations break down the approach used provides a general framework for the analysis and interpretation of numerical instability in approximations of dissipative nonlinear partial differential equations continuous and discrete problems are studied from the perspective of bifurcation theory and numerical instability is shown to be associated with the bifurcation of periodic orbits in discrete systems an asymptotic approach due to newell siam j appl math 33 1977 133 160 is used to investigate the instability phenomenon further in particular equations are derived that describe the interaction of the dynamics of the partial differential equation with the artefacts of the discretization

on spurious fixed points of runge kutta methods in this paper we investigate the onset of spurious fixed points when runge kutta methods are applied to study the dynamics of differential equations it is shown computationally that the spurious equilibria of griffithset al 14 are connected at infinity with fixed points inherited from the differential equation we introduce and study the concept of b regularity which is in connection to the concept of regularity introduced by iserles principium cuius hinc nobis exordia sumet nullam rem e nilo gigni diuinitus umquam nam si de nilo fierent ex omnibus rebus omne genus nasci posset nil semine egeret de rerum natura liber primus 149 lucrecio

nonlinear diffusion and stable period 2 solutions of a discrete reaction diffusion model abstract a discrete form of a nonlinear parabolic equation is considered this equation is in fact a model for migrating populations and contains a quadratic reaction term and a diffusion term which can be either linear for nonlinear attention is focused on the case of nonlinera diffusion and in particular period 2 solutions of the discrete version conditions for the existence and stability of two types of period 2 solutions are determined analytically these analytical results which are local in the neighborhood of fixed points are compared with the results of some numerical experiments showing that one of these period 2 solutions is a considerably stronger attractor than the other