harmonic maps and periodic toda systems harmonic maps from the point of view of periodic toda systems are considered the correspondence between certain harmonic immersions of surfaces and solutions to periodic toda systems related to nonexceptional simple lie algebras is constructed

harmonic maps into the orthogonal group and null curves we find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group mathrm o n in terms of free holomorphic data which lead to formulae for all such harmonic maps our work reveals an interesting correspondence between certain harmonic maps and the free weierstrass representation of null curves and minimal surfaces in 3 and 4 space

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

new criteria for ergodicity and nonuniform hyperbolicity in this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected riemannian manifolds this method allows us to give a more accurate description of certain ergodic components the use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero lyapunov exponents in some contexts in the partial hyperbolicity context we obtain that stably ergodic diffeomorphisms are c1 dense among volume preserving partially hyperbolic diffeomorphisms with 2 dimensional center bundle this is motivated by a well known conjecture of pugh and shub

pseudo holomorphic curves in complex grassmann manifolds it is proved that the kahler angle of the pseudo holomorphic sphere of constant curvature in complex grassmannians is constant at the same time we also prove several pinching theorems for the curvature and the kahler angle of the pseudo holomorphic spheres in complex grassmannians with non degenerate associated harmonic sequence

stability analysis with popov multipliers and integral quadratic constraints it is shown that a general form of popov multipliers can be used in stability analysis based on integral quadratic constraints iqc the popov multiplier is nonproper and a condition that the nominal plant is strictly proper will be imposed in order to ensure boundedness of the iqc corresponding to the popov multiplier a consequence of our main result is that the classical popov criterion can be combined with a stability criterion for slope restricted nonlinearities developed by zames and falb an example shows that the combination of these two criteria is useful in applications

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

semi continuity of the entropy of sinai probability measures for partially hyperbolic diffeomorphisms abstract we establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples

the unified approach to integrable relativistic equations soliton solutions over nonvanishing backgrounds ii in part i of this work the n soliton solution has been constructed for the generic system associated with the sl 2 c case of the scheme for the unified description of integrable relativistic massive fields here solutions are isolated for reductions of this system including the conventional complex sine gordon equation the massive thirring model and another complexification of the sine gordon equation defined by the lagrangian l 2 1 2 2 j 2 2 2 1 2 j i the latter model is shown to exhibit decays and fusion of subluminal solitons the reduction to the conventional complex sine gordon appears to be even more interesting as it cannot be defined by simply restricting the linear problem to some real form of sl 2 c algebra and the relevant involution turns out to be quite nontrivial when the background is flat this involution degenerates and so the n kink solution for the reduction cannot be extracted from the generic n kink solution directly this difficulty is

vertex operator representation of some quantum tori lie algebras we are defining the trigonometric lie subalgebras in bar x infty bar a infty bar b infty bar c infty bar d infty which are the natural generalization of the well known sin lie algebra the embedding formulas into bar x infty are introduced these algebras can be considered as some lie algebras of quantum tori an irreducible representation ofa b series of trigonometric lie algebras is constructed special cases of the trigonometric lie factor algebras which can be considered as a quantum preserving lie algebra structure deformation of the kac moody algebras are considered

dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to anosov we show that partially hyperbolic diffeomorphisms of d dimensional tori isotopic to an anosov diffeomorphism where the isotopy is contained in the set of partially hyperbolic diffeomorphisms are dynamically coherent moreover we show a global stability result i e every partially hyperbolic diffeomorphism as above is leaf conjugate to the linear one as a consequence we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one dimensional center direction that are isotopic to anosov diffeomorphisms through such a path

quantum line operators from lax pairs motivated by the realization of the yang baxter equation of 2d integrable models in the 4d gauge theory of costello witten yamazaki cwy we study the embedding of integrable 2d toda field models inside this construction this is done by using the lax formulation of 2d integrable systems and by thinking of the standard lax pair l in terms of components of cwy gauge connection propagating along particular directions in the gauge bundle we also use the results of the cwy theory to build quantum line operators for 2d toda models and compute the one loop contribution of two intersecting lines exchanging one gluon other features such as local symmetries and comments on extension of our method to other 2d integrable models are also discussed we also comment on some basic points that still need a refinement before discussing about a fully consistent embedding of lax pairs into cwy theory

theory of group representations and integration of nonlinear systems xa zz exp kx a abstract a constructive proof of complete integrability of the system xa zf z exp kx a and its explicit solution are given for k coinciding with the cartan matrix of simple lie algebras the technique for dealing with this system of essentially nonlinear two dimensional partial differential equations involves lax type representation and representation theory of the corresponding group

geometric and measure theoretical structures of maps with mostly contracting center we show that every diffeomorphism with mostly contracting center direction exhibits a geometric combinatorial structure which we call skeleton that determines the number basins and supports of the physical measures furthermore the skeleton allows us to describe how the physical measures bifurcate as the diffeomorphism changes in particular we use this to construct examples with any given number of physical measures with basins densely intermingled and to analyse how these measures collapse into each other through explosions of their basins as the dynamics varies this theory also allows us to prove that in the absence of collapses the basins are continuous functions of the diffeomorphism

classical and quantum integrable systems with boundaries we study two dimensional classically integrable field theory with independent boundary conditions at each end and obtain three possible generating functions for integrals of motion when this model is an ultralocal one classically integrable boundary conditions can be found when solving boundary k equations in the quantum case we also find that the unitarity condition of the quantum r matrix is sufficient to construct commutative quantities with boundaries and its reflection equations are obtained

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

totally real submanifolds in s 6 1 satisfying chen s equality in this paper we study 3 dimensional totally real submanifolds of s6 1 if this submanifold is contained in some 5 dimensional totally geodesic s5 1 then we classify such submanifolds in terms of complex curves in cp 2 4 lifted via the hopf fibration s5 1 cp 2 4 we also show that such submanifolds always satisfy chen s equality i e m 2 where m p p inf k p for every p m then we consider 3 dimensional totally real submanifolds which are linearly full in s6 1 and which satisfy chen s equality we classify such submanifolds as tubes of radius 2 in the direction of the second normal space over an almost complex curve in s6 1

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

the space of almost complex 2 spheres in the 6 sphere the complex dimension of the space of linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is calculated to be d 8 explicit examples of these objects are constructed for every integer value of the degree d 6 d 6 7 furthermore it is shown that when d 6 this space is isomorphic to the group g2 c and when d 7 this space is empty we also show that the dimension of the space of non linearly full almost complex 2 spheres of area 4 d in the round 6 sphere is 2d 5

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

a note on affine toda couplings the author proves the clebsch gordan property for the couplings of an untwisted affine toda field theory utilizing dorey s 1991 conjecture

two dimensional generalized toda lattice the zero curvature representation is obtained for the two dimensional generalized toda lattices connected with semisimple lie algebras the reduction group and conservation laws are found and the mass spectrum is calculated

accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d center bundle we prove that stable ergodicity is cr open and dense among conservative partially hyperbolic diffeomorphisms with one dimensional center bundle for all r 2

on the bernoulli property for certain partially hyperbolic diffeomorphisms abstract we address the classical problem of equivalence between kolmogorov and bernoulli property of smooth dynamical systems in a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to anosov derived from anosov on 3 torus we prove that kolmogorov and bernoulli properties are equivalent in our approach we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms as an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from anosov diffeomorphism of 3 torus and its linearization our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi isometric

horseshoes for diffeomorphisms preserving hyperbolic measures we give extensions of katok s horseshoe constructions comment on related results and provide a self contained proof we consider either a c 1 alpha diffeomorphism preserving a hyperbolic measure or a c 1 diffeomorphism preserving a hyperbolic measure whose support admits a dominated splitting

more non abelian loop toda solitons we find new solutions including soliton like ones for a special case of non abelian loop toda equations associated with complex general linear groups we use the method of rational dressing based on an appropriate block matrix representation suggested by the gradation under consideration we present solutions in the form of a direct matrix generalization of hirota s soliton solution already well known in the case of abelian loop toda systems

technical communique lyapunov functions for the multivariable popov criterion with indefinite multipliers we demonstrate the existence of lur e postnikov type lyapunov functions for a reasonably general form of the multivariable popov criterion in particular the multipliers need not be positive semi definite we discuss the application to the special cases where the feedback element is either diagonal unstructured or linear

a volume preserving diffeomorphism with essential coexistence of zero and nonzero lyapunov exponents we show that there exists a c volume preserving topologically transitive diffeomorphism of a compact smooth riemannian manifold which is ergodic indeed is bernoulli on an open and dense subset mathcal g of not full volume and has zero lyapunov exponent on the complement of mathcal g

kac moody groups and integrability of soliton equations a new approach to integrability of affine toda field theories and closely related to them kdv hierarchies is proposed the flows of a hierarchy are explicity identified with infinitesimal action of the principal abelian subalgebra of the corresponding affine kac moody algebra on a homogeneous space

local conservation laws for the two dimensional periodic su n 1 toda lattices an infinite number of local conservation laws are derived for the two dimensional periodic su n 1 toda lattice equations following the well known method for the sine gordon equation

all harmonic 2 spheres in the unitary group completely explicitly we give a completely explicit formula for all harmonic maps of finite uniton number from a riemann surface to the unitary group u n in any dimension and so all harmonic maps from the 2 sphere in terms of freely chosen meromorphic functions on the surface and their derivatives using only combinations of projections and avoiding the usual bar partial problems or loop group factorizations we interpret our constructions using segal s grassmannian model giving an explicit factorization of the algebraic loop group and showing how to obtain harmonic maps into a grassmannian

on the squared eigenfunction symmetry of the toda lattice hierarchy the squared eigenfunction symmetry for the toda lattice hierarchy is explicitly constructed in the form of the kronecker product of the vector eigenfunction and the vector adjoint eigenfunction which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function respectively then after the fay like identities and some important relations about the wave functions are investigated the action of the squared eigenfunction related to the additional symmetry on the tau function is derived which is equivalent to the adler shiota van moerbeke formulas

a popov criterion for systems with slowly time varying parameters a popov criterion is derived for systems with slowly time varying parameters the parameters and their time derivatives are assumed to take values in convex polytopes

classical a n w geometry by analyzing theextrinsic geometry of two dimensional surfaces chirally embedded incp n thecp n w surface 1 we give exact treatments in various aspects of the classical w geometry in the conformal gauge first the basis of tangent and normal vectors are defined at regular points of the surface such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the conformal a n toda lax pair since the latter is known to be intrinsically related with the w symmetries this gives the geometrical meaning of thea n w algebra second w surfaces are put in one to one correspondence with solutions of the conformally reduced wznw model which is such that the toda fields give the cartan part in the gauss decomposition of its solutions third the additional variables of the toda hierarchy are used as coordinates ofcp n this allows us to show that w transformations may be extended as particular diffeomorphisms of this target space higher dimensional generalizations of the wznw equations are derived and related with the zakharov shabat equations of the toda hierarchy fourth singular points are studied from a global viewpoint using our earlier observation 1 that w surfaces may be regarded as instantons the global indices of the w geometry which are written in terms of the toda fields are shown to be the instanton numbers for associated mappings of w surfaces into the grassmannians the relation with the singularities of w surface is derived by combining the toda equations with the gauss bonnet theorem

essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms we prove that any diffeomorphism of a compact manifold can be approximated in the c 1 topology by another diffeomorphism exhibiting a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by one which is essentially hyperbolic has a finite number of transitive hyperbolic attractors with open and dense basin of attraction

non crystallographic reduction of generalized calogero moser models we apply a recently introduced reduction procedure based on the embedding of non crystallographic coxeter groups into crystallographic ones to calogero moser systems for rational potentials the familiar generalized calogero hamiltonian is recovered for the hamiltonians of trigonometric hyperbolic and elliptic types we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio we explicitly show for the simplest of these non crystallographic models how the corresponding classical equations of motion can be derived from a lie algebraic lax pair based on the larger crystallographic coxeter group

linearizing integral transform for the multicomponent lattice kp abstract we present a generalization of the direct linearization method for the kadomtsev petviashvili kp equation and its lattice analogues such as the two dimensional toda equation the generalization consists of a riemann hilbert type of integral transform which relates solutions of the associated spectral problem to any given solution of the spectral problem by dimensional reduction we obtain the integral transform for the korteweg de vries kdv equation and its lattice analogues the integral transform is well suited for the investigation of the finite matrix generalizations of the above mentioned equations such finite matrix generalizations may be of use in connection with partial difference equations for spin correlation functions in the two dimensional ising model

toda lattice mass transport in lagrangian mechanics and in a two dimensional system abstract this paper explores the connection between the hydrodynamic mass transport description and the thermodynamic description for a nonlinear range of the toda lattices particular attention is paid to the broken isotropy in the kdv and burgers equations the flow variable representation is established from the lagrangian mechanics for hydrodynamic mass transport based on the inverse scattering transform the gel fand levitan marchenko glm equation is formulated from the kdv equation expressed by the flow variable representation we found that a kernel of the glm equation is given by the concentration variable q x t a lagrangian is formulated for the kdv equation in state space q x t k x t next an extension of the flow variable representation is sought in a two dimensional system the lhs of the kadomtsev petviashvili kp equation takes the same form as in the second formalism of the kdv equation by setting up the flow variable representation of the kp equation the burgers equation in two dimensions is formulated these results contribute to an understanding of the broken isotropy for the nonlinear mass transport equations these results provide physical insight into various consequences of the generalized form of the kawasaki ohta equation from the viewpoint of mass transport

tournaments flags and harmonic maps twistor space constructions have shown that under quite general hypotheses harmonic maps of riemann surfaces into riemannian manifolds can be represented by holomorphic curves with respect to almost complex structures which are often non integrable see for example 19 17 5 for maps into complex grassmannians the almost complex structures are invariant ones on flag manifolds and can be identified with tournaments i e digraphs in which any two nodes are joined by exactly one oriented edge elementary properties of tournaments are summarised in sect i the basic references are moon 14 and reid and beineke 18 emphasis is given to certain types of tournament which are interpreted in sect 2 in terms of the homogeneous structure of complex flag manifolds holomorphic curves in such flag manifolds are described in sect 3 by collections of subbundles of a trivial bundle over a riemann surface whose second fundamental forms define a tournament this approach relies on techniques of burstall and wood 6 and gives rise naturally to harmonic maps into grassmannians when the riemann surface has genus zero a result from algebraic geometry namely the birkhoff grothendieck decomposition theorem for holomorphic vector bundles fits into the scheme and allows one to reconstruct a tournament from a harmonic map into a grassmannian in sect 4 we use such techniques to show that harmonic maps of tl2p 1 into a complex grassmannian can be built up successively from holomorphic data this last result extends results of burstall and wood 6 and is closely related to recent work of uhlenbeck 21 and wolfson 23

the determinant representation for quantum correlation functions of the sinh gordon model we consider the quantum sinh gordon model in this paper form factors in this model were calculated by mussardo and colleagues we sum up all contributions of form factors and obtain a closed expression for a correlation function this expression is a determinant of an integral operator similar determinant representations have been proven to be useful not only in the theory of correlation functions but also in matrix models

real robustness bounds using generalized stability multipliers sufficient conditions for robust stability using generalized stability multipliers are presented for systems with sector and norm bounded block structured real uncertainty two parametrizations of the multiplier are considered and the robustness criteria are written as linear matrix inequalities upper bounds for the peak structured singular value over frequency which eliminate frequency gridding are then derived numerical examples provide a comparison of the peak upper bounds obtained using the two multiplier parametrizations these examples show that the conservatism of the peak upper bounds is reduced by increasing the dynamic order of the multiplier

extremal lyapunov exponents an invariance principle and applications we propose a new approach to analyzing dynamical systems that combine hyperbolic and non hyperbolic center behavior e g partially hyperbolic diffeomorphisms a number of applications illustrate its power

sequences of willmore surfaces in this paper we develop the theory of willmore sequences for willmore surfaces in the 4 sphere we show that under appropriate conditions this sequence has to terminate in this case the willmore surface either is the twistor projection of a holomorphic curve into mathbb c mathbb p 3 or the inversion of a minimal surface with planar ends in mathbb r 4 these results give a unified explanation of previous work on the characterization of willmore spheres and willmore tori with non trivial normal bundles by various authors

on the ergodicity of partially hyperbolic systems pugh and shub have conjectured that essential accessibility implies ergodicity for a c partially hyperbolic volume preserving diffeomorphism we prove this conjecture under a mild center bunching assumption which is satisfied in particular by all partially hyperbolic systems with 1 dimensional center bundle we also obtain ergodicity results for c 1ci partially hyperbolic systems

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse

a popov criterion for uncertain linear multivariable systems the popov absolute stability criterion is traditionally proved using a lyapunov function and the positive real lemma in this paper a simplified proof of the multivariable popov criterion is given for the case of one sided sector bounded real parameter uncertainty a loop shifting transformation is then used to extend the popov criterion to two sided sector bounded uncertain matrices specialization of this result to norm bounded uncertain matrices leads to an upper bound for the structured singular value for block structured real parameter uncertainty

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

almost all cocycles over any hyperbolic system have nonvanishing lyapunov exponents we prove that for any s 0 the majority of c s linear cocycles over any hyperbolic uniformly or not ergodic transformation exhibit some nonzero lyapunov exponent this is true for an open dense subset of cocycles and actually vanishing lyapunov exponents correspond to codimension 1 this open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

bethe ansatz equations for the classical a 1 _ n affine toda field theories we establish a correspondence between classical affine toda field theories and an bethe ansatz systems we show that the connection coefficients relating specific solutions of the associated classical linear problem satisfy functional relations of the type that appear in the context of the massive quantum integrable model

real m bounds based on fixed shapes in the nyquist plane parabolas hyperbolas cissoids nephroids and octomorphs in this paper we introduce new bounds for the real structured singular value the approach is based on absolute stability criteria with plant dependent multipliers that exclude the nyquist plot from fixed plane curve shapes containing the critical point jo unlike half plane and circle based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty since the plant dependent multipliers have the same functional form at all frequencies the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency dependent off axis circles that arise in standard real bounds

asymptotic measure expansive diffeomorphisms abstract in this paper the notion of asymptotic measure expansiveness is introduced and its relationship with dominated splitting is considered it is proved that if a diffeomorphism admits a co dimension one dominated splitting then it is asymptotic measure expansive also a diffeomorphism with a homoclinic tangency can be perturbed to a non asymptotic measure expansive diffeomorphism