unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

fredholm differential operators with unbounded coefficients abstract we prove that a first order linear differential operator g with unbounded operator coefficients is fredholm on spaces of functions on r with values in a reflexive banach space if and only if the corresponding strongly continuous evolution family has exponential dichotomies on both r and r and a pair of the ranges of the dichotomy projections is fredholm and that the fredholm index of g is equal to the fredholm index of the pair the operator g is the generator of the evolution semigroup associated with the evolution family in the case when the evolution family is the propagator of a well posed differential equation u t a t u t with generally unbounded operators a t t r the operator g is a closure of the operator d dt a t thus this paper provides a complete infinite dimensional generalization of well known finite dimensional results by palmer and by ben artzi and gohberg

polynomial growth rates abstract we consider linear equations v a t v with a polynomial asymptotic behavior that can be stable unstable and central we show that this behavior is exhibited by a large class of differential equations by giving necessary and sufficient conditions in terms of generalized polynomial lyapunov exponents for the existence of polynomial behavior in particular any linear equation in block form in a finite dimensional space with three blocks having polynomial lyapunov exponents respectively negative positive and zero has a nonuniform version of polynomial trichotomy which corresponds to the usual notion of trichotomy but now with polynomial growth rates we also obtain sharp bounds for the constants in the notion of polynomial trichotomy in addition we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction

an ergodic theorem in the qualitative behavior of non linear semiflows abstract take to be a continuous semiflow on the locally compact metric space and let a be a family of possibly unbounded densely defined closed operators on the banach space x we show that the solutions of the variational equation u a t u t r have a uniform exponential decay if and only if there exist p 1 and a sequence of positive real numbers n n n with n satisfying lim n n n 0 n x p d 0 uniformly w r t for all x x where t t is a strongly continuous cocycle solving in the classical sense the above variational equation thus we generalize some known results obtained by datko and pazy

limit distributions of directionally reinforced random walks abstract as a mathematical model for time and space correlations in ocean surface wave fields mauldin monticino and von weizsacker 1996 adv in math 117 239 252 introduced directionally reinforced random walks and analyzed their recurrence properties this paper develops limiting properties for these random walks inrd in particular it is shown that the limit distribution is a brownian motion if the time between changes of directions has a finite second moment and is a stable process if the time between changes of directions is in a domain of attraction of a stable law the latter gives an affirmative answer to an open question posed by mauldinet al the strong law of large numbers and strong approximations are also discussed

nonuniform spectrum on banach spaces abstract for a sequence of bounded linear operators acting on a banach space we consider the notion of nonuniform spectrum this is defined in terms of the existence of nonuniform exponential dichotomies with an arbitrarily small nonuniform part and can be seen as a nonuniform version of the spectrum introduced by sacker and sell in the case of a single trajectory although now in the infinite dimensional setting we give a complete characterization of all possible forms of the nonuniform spectrum for sequences of compact linear operators and more generally for sequences of bounded linear operators satisfying a certain asymptotic compactness moreover we provide explicit examples of sequences of compact linear operators acting on the l 2 space of sequences of real numbers for all the possible forms of the nonuniform spectrum as nontrivial applications we show that the nonuniform spectrum of a lyapunov regular sequence is the set of lyapunov exponents and that the asymptotic behavior persists under sufficiently small nonlinear perturbations in the sense that the lower and upper lyapunov exponents of the perturbed dynamics belong to a connected component of the nonuniform spectrum finally we obtain appropriate versions of the results for nonuniformly hyperbolic cocycles

invariance principles for renewal processes on presente une methodologie generale pour demontrer des principes d invariance pour des processus de renouvellement on obtient des approximations presque sures et d inegalites de probabilites

holomorphic isometries of mathbb b m into bounded symmetric domains arising from linear sections of minimal embeddings of their compact duals we study general properties of images of holomorphic isometric embeddings of complex unit balls mathbb b m into irreducible bounded symmetric domains varomega of rank at least 2 in particular we show that such holomorphic isometries with the minimal normalizing constant arise from linear sections varlambda of the compact dual x c of varomega the question naturally arises as to which linear sections z varlambda cap varomega are actually images of holomorphic isometries of complex unit balls we study the latter question in the case of bounded symmetric domains varomega of type iv alias lie balls i e bounded symmetric domains dual to hyperquadrics we completely classify images of all holomorphic isometric embeddings of complex unit balls into such bounded symmetric domains varomega especially we show that there exist holomorphic isometric embeddings of complex unit balls of codimension 1 incongruent to the examples constructed by mok proc am math soc 144 4515 4525 2016 from varieties of minimal rational tangents and that moreover any holomorphic isometric embedding f mathbb b m rightarrow varomega extends to a holomorphic isometric embedding f mathbb b n 1 rightarrow varomega dim varomega n the case of lie balls is particularly relevant because holomorphic isometric embeddings of complex unit balls of sufficiently large dimensions into an irreducible bounded symmetric domain other than a type iv domain are expected to be more rigid

a strong convergence to the rosenblatt process we give a strong approximation of rosenblatt process via transport processes and we give the rate of convergence

on the rate of convergence to rosenblatt type distribution the main result of the article is the rate of convergence to the rosenblatt type distributions in non central limit theorems specifications of the main theorem are discussed for several scenarios in particular special attention is paid to the cauchy generalized linnik s and local global distinguisher random processes and fields direct analytical methods are used to investigate the rate of convergence in the uniform metric

markus yamabe conjecture for non autonomous dynamical systems abstract the aim of this paper is the study of the problem of global asymptotic stability of trivial solutions of non autonomous dynamical systems both with continuous and discrete time we study this problem in the framework of general non autonomous dynamical systems cocycles in particular we present some new results for a non autonomous version of the markus yamabe conjecture

smooth stable invariant manifolds and arbitrary growth rates abstract for nonautonomous linear equations v a t v with a generalized exponential dichotomy we show that there is a smooth stable invariant manifold for the perturbed equation v a t v f t v provided that f is sufficiently small the generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates e c t for some function t we consider the general case of nonuniform exponential dichotomies and the result is obtained in banach spaces moreover we show that for an equivariant system the dynamics on the stable manifold in a certain class of graphs is also equivariant we emphasize that this result cannot be obtained by averaging over the symmetry

almost periodic solutions of lasota wazewska type delay differential equation sufficient conditions are derived for the existence of a globally attractive almost periodic solution of a single species model given by the nonautonomous lasota wazewska type delay differential equations x t txt ptfxt in which 0 p t 0 t are continuous almost periodic functions and f is a decreasing positive c1 function conditions of the uniform asymptotical stability for some associated nonautonomous delay differential equations x t txt ptxt are also discussed

volume preserving mappings between hermitian symmetric spaces of compact type abstract in this paper we establish the rigidity result for local holomorphic volume preserving maps from an irreducible hermitian symmetric space of compact type into its cartesian products

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

lp r x lq r x admissibility and exponential dichotomies of cocycles we analyze the existence of eventually no past exponential and ordinary dichotomies of an exponentially bounded strongly continuous cocycle t t r over a continuous semiflow our main tool is the lp r x lq r x admissibility condition i e there exist p q 1 p q 1 such that for each input f lp r x and there exists x x such that the output u x f r x u t x f t x 0t s t s f s ds belongs to lq r x we prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to requiring that the boundedness to be uniform with respect to we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for t t r variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained it is worth to note that we involve a concept of a no past exponential dichotomy for cocycles weaker than the well known concept defined by sacker and sell 1994 33 our definition of exponential dichotomy follows partially the definition given in chow and leiva 1996 2 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to perron 1930 23 daleckij and krein 1974 6 massera and schaffer 1966 14 nguyen van minh rabiger and r schnaubelt 1998 18

existence of nonuniform exponential dichotomies and a fredholm alternative abstract we consider linear equations x a t x in a banach space with nonuniform exponential dichotomies in r and r and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations we also obtain a fredholm alternative for the existence of bounded solutions under nonlinear perturbations

the complex brownian motion as a weak limit of processes constructed from a poisson process in this paper we show an approximation in law of the complex brownian motion by processes constructed from a unique standard poisson process

long time behavior of almost periodically forced parabolic equations on the circle abstract we study the long time behavior of the skew product semiflow generated by scalar reaction diffusion equation on the circle with almost periodic forcing u t u x x f t u u x t 0 x s 1 r 2 z where f t u u x is uniformly almost periodic in t almost periodic environmental forcing exhibits the external effects which are roughly but not exactly periodic contrary to the time periodic cases for which any limit set can be embedded into a periodically forced circle flow we show that for almost periodic forcing the problem that whether can be embedded into an almost periodically forced circle flow is strongly related to the dimension of the center space v c associated with on the one hand if dim v c 1 then is either spatially inhomogeneous or spatially homogeneous and moreover any spatially inhomogeneous can be embedded into an almost periodically forced circle flow on the other hand when dim v c 1 it is shown that the above embedding property cannot hold anymore these reveal that for such system there are essential differences between time periodic forcing and non periodic forcing

the generic gradient like structure of certain asymptotically autonomous semilinear parabolic equations abstract we consider asymptotically autonomous semilinear parabolic equations u t a u f t u suppose that f t f as t where the semiflows induced by u t a u f u are gradient like under certain assumptions it is shown that generically with respect to a perturbation g with g t 0 as t every solution of u t a u f t u g t is a connection between equilibria e of with m e m e moreover if the morse indices satisfy m e m e then u is isolated by linearization

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

attractor minimal sets for cooperative and strongly convex delay differential systems abstract we study the skew product semiflow induced by a family of convex and cooperative delay differential systems under some monotonicity assumptions we obtain an ergodic representation for the upper lyapunov exponent of a minimal subset in addition when eventually strong convexity at one point is assumed and there exist two completely strongly ordered minimal subsets k1 ck2 we show that k1 is an attractor subset which is a copy of the base the long time behaviour of every trajectory strongly ordered with k2 is then deduced some examples of application of the theory are shown

approximating lyapunov exponents and sacker sell spectrum for retarded functional differential equations we consider lyapunov exponents and sacker sell spectrum for linear nonautonomous retarded functional differential equations posed on an appropriate hilbert space a numerical method is proposed to approximate such quantities based on the reduction to finite dimension of the evolution family associated to the system to which a classic discrete qr method is then applied the discretization of the evolution family is accomplished by a combination of collocation and generalized fourier projection a rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra the efficacy of the results is illustrated with some numerical examples

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

an extension of a theorem of r datko to the case of non uniform exponential stability of linear skew product semiflows abstract in 1970 r datko proved that the trajectories of a c 0 semigroup t t t 0 on a hilbert space x exhibit an exponential decay if and only if they stay in l 2 r x datko s theorem was crucial in extending the lyapunov operator equation to the case of autonomous systems x a x with unbounded a extensions have been done to the case of uniform exponential stability of evolution families and more recently of lsps linear skew product semiflow the aim of the present paper is to extend datko s result to the general case of non uniform exponential stability of the lsps that does not necessarily possess a uniform exponential growth

variations and estimators for self similarity parameters via malliavin calculus using multiple stochastic integrals and the malliavin calculus we analyze the asymptotic behavior of quadratic variations for a specific non gaussian self similar process the rosenblatt process we apply our results to the design of strongly consistent statistical estimators for the self similarity parameter h although in the case of the rosenblatt process our estimator has non gaussian asymptotics for all h 1 2 we show the remarkable fact that the process s data at time 1 can be used to construct a distinct compensated estimator with gaussian asymptotics for h in 1 2 2 3

global attractivity in concave or sublinear monotone infinite delay differential equations abstract we study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities we analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way for which we prove the existence of a globally asymptotically stable minimal set given by a 1 cover of the base flow we apply these results to the description of the long term dynamics of a nonautonomous model representing a stage structured population growth without irreducibility assumptions on the coefficient matrices

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

on holomorphic isometric immersions of nonhomogeneous kahler einstein manifolds into the infinite dimensional complex projective space abstract in the paper we study the existence of holomorphic isometric immersions from nonhomogeneous kahler einstein manifolds into infinite dimensional complex projective space it can also be regarded as an application of explicit solutions of complex monge ampere equations on some pseudoconvex domains

nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces abstract in this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew product flows in infinite dimensional spaces we treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families we show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary our results generalize the previous studies on this topic and provide as particular cases many interesting situations among which we mention the detection of the exponential trichotomy of general non autonomous systems

on the asymptotic behavior of discrete dynamical systems an ergodic theory approach abstract the aim of this paper is to present a new and general method for the study of stability expansiveness and dichotomy of dynamical systems based on arguments proceeding from ergodic theory and on admissibility properties we consider a discrete variational system and we associate to it three different families of input output systems corresponding to the investigations for stability expansiveness and dichotomy respectively we introduce some input output conditions relative to ergodic measures and we show that certain nonuniform admissibility properties imply the existence of the uniform asymptotic behaviors first we obtain characterizations for uniform exponential stability as an application we prove that in order to preserve the uniform exponential stability subject to perturbations it is sufficient for the perturbed system to be close enough to the initial system on a measurable set next we deduce criteria for uniform exponential expansiveness based on some exact admissibilities relative to ergodic measures finally we present an interesting application to the study of the uniform exponential dichotomy starting from a nonuniform admissibility property and using stability criteria

boundedness global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations abstract the paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations we carry out continuation of local solutions to maximal solutions in a functional analytic setting for maximal solutions we establish global existence via exponential boundedness and by a contraction argument adapted to derive uniform existence time moreover within the setting of dual banach spaces we derive results on continuous dependence with respect to time and initial state to achieve generality the paper is organized top down in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold

spectral theory for general nonautonomous random dispersal evolution operators abstract we investigate the spectral theory of the following general nonautonomous evolution equation t u t x a u t x h t x u t x x d where d is a bounded subset of r n which can be a smooth domain or a discrete set a is a general linear dispersal operator for example a laplacian operator an integral operator with positive kernel or a cooperative discrete operator and h t x is a smooth function on r d we first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal lyapunov exponent of a time dependent dispersal equation is always greater than or equal to that of the time averaged one several results about the principal eigenvalue of time periodic parabolic equations are extended to general time periodic dispersal ones finally the investigation is generalized to random time dependent dispersal equations

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

exponential trichotomy and p admissibility for evolution families on the real line the aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line we prove that if p 1 and the pair c b r x c c r x is uniformly p admissible for an evolution family open image in new window u t s t s then open image in new window is uniformly exponentially trichotomic after that we analyze when the uniform p admissibility of the pair c b r x c c r x becomes a necessary condition for uniform exponential trichotomy as applications of these results we study the uniform exponential dichotomy of evolution families we obtain that in certain conditions the admissibility of the pair c b r x l p r x for an evolution family open image in new window u t s t s is equivalent with its uniform exponential dichotomy

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

towards a perron frobenius theory for eventually positive operators abstract this article is a contribution to the spectral theory of so called eventually positive operators i e operators t which may not be positive but whose powers t n become positive for large enough n while the spectral theory of such operators is well understood in finite dimensions the infinite dimensional case has received much less attention in the literature we show that several sensible notions of eventual positivity can be defined in the infinite dimensional setting and in contrast to the finite dimensional case those notions do not in general coincide we then prove a variety of typical perron frobenius type results we show that the spectral radius of an eventually positive operator is contained in the spectrum we give sufficient conditions for the spectral radius to be an eigenvalue admitting a positive eigenvector and we show that the peripheral spectrum of an eventually positive operator is a cyclic set under quite general assumptions all our results are formulated for operators on banach lattices and many of them do not impose any compactness assumptions on the operator

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

rate of convergence of transport processes with an application to stochastic differential equations we obtain a rate of convergence of uniform transport processes to brownian motion which we apply to the wong and zakai approximation of stochastic integrals

on lipschitz dependence in systems with differentiated inputs where a e r d f and gi i 1 k are vector fields on r and z 1 z e c1 0 lj doss 21 and sussmann 17 have shown that under appropriate conditions on the coefficients f and g i 1 k see 2 below the mapping s which takes z z z k into the solution x of l admits a continuous w r to the sup norm extension to c 0 1 r if one wants to extend the mapping s by continuity to inputs say with jumps an lp norm rather than the sup norm is suitable in contrary to what is asserted in 13 lemma 2 and theorem 1 we show example 1 that under the assumptions 2 in general s does not admit a continuous extension to l 1 rk s is however lipschitz continuous with respect to lp norm on each l set theorem 1 hence admits a continuous extension to l 0 1 r k in this sense viewed as a mapping from d 0 1 r k the space of right continuous functions with left limits into d 0 1 d s will be shown to be continuous w r to the skorokhod topology 1 7 and local lipschitz dependence in the skorokhod metrics will be discussed theorem 2 for z e d 0 1 r k with finite quadratic variation x s z obeys an integral eq 4 which is a deterministic analogue of the stochastic integral eq 9 in 13 there called canonical extension of 1 for semimartingale inputs it is obvious from 9 that the mapping s gives a pathwise solution of a certain stochastic differential equation if z z co is e g a semimartingale especially ff z

strong nonuniform behaviour a datko type characterization abstract we obtain a datko type characterization of several classes of a strong nonuniform exponential behaviour we consider separately the case of exponential contractions expansions and dichotomies in addition we consider both discrete and continuous time as a nontrivial application of our work we study the robustness property of strong nonuniform exponential contractions expansions and dichotomies

averaging theorems for the large time behavior of the solutions of nonautonomous systems abstract we obtain some averaging theorems for the large time behavior of an evolution family u t s t s 0 acting on a banach space it is known that if a trajectory u t 0 x 0 is asymptotically stable then its p mean tends to zero we will show here that if the uniformly weighted p means of all the trajectories starting on the unit sphere are bounded then u t s t s 0 is uniformly exponentially stable while the converse statement is a simple verification discrete time versions of this result are given also variants for the uniform exponential blow up are obtained thus we generalize some known results obtained by r datko a pazy and v pata

exponential stability of operators and operator semigroups extending earlier results of datko pazy and littman on c0 semigroups and of przyluski and weiss on operators we prove the following let t be a bounded linear operator on a banach space x and let r t denotes its spectral radius let e be a banach function space over n with the property that limn 0 n 1 e if for each x x and x x the map n x tnx belongs to e then r t 0 for all t 0 if n 0 x tnx for all x x 1 then r t 1 assuming a 2 condition on a further improvement is obtained for locally bounded semigroups t t t t 0 we obtain similar results in terms of the maps t t t x

rigidity for local holomorphic isometric embeddings from mathbb b n into mathbb b n_1 mathbb b n_m up to conformal factors in this article we study local holomorphic isometric embeddings from mathbb b n into mathbb b n 1 mathbb b n m with respect to the normalized bergman metrics up to conformal factors assume that each conformal factor is smooth nash algebraic then each component of the map is a multi valued holomorphic map between complex euclidean spaces by the algebraic extension theorem derived along the lines of mok and mok and ng applying holomorphic continuation and analyzing real analytic subvarieties carefully we show that each component is either a constant map or a proper holomorphic map between balls applying a linearity criterion of huang we conclude the total geodesy of non constant components

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

meagre functions and asymptotic behaviour of dynamical systems a measurable function x j r x x a metric space is said to be c meagre if c x is non empty and for every closed set k x with k c x 1 k has 5nite lebesgue measure this concept of meagreness is shown to provide a unifying framework which facilitates a variety of characterizations extensions or generalizations of diverse facts pertaining to asymptotic behaviour of dynamical systems by way of motivation consider the initial value problem f x x 0 rn with f rn rn locally lipschitz and denote its unique solution by x t t de5ned on its maximal interval of existence i 0 let g rn r be continuous assume that x is a bounded solution of the initial value problem in which case i 0 then g x is uniformly continuous if in addition g x l1 then we may conclude by an elementary observation frequently referred to as barb alat s lemma 5 see also 18 section 21 lemma 1 that g x t 0 as t 0 since by boundedness x t approaches as t its non empty compact invariant limit set b we also conclude that b g 1 0 it follows that x t tends as t to the largest invariant subset of g 1 0 this is a short proof of theorem 1 2 of 6 the proof in 6 is based on arguments involving properties of the dow generated by the dieerential

on weak exponential stability of evolution operators in banach spaces abstract the aim of this paper is to give some characterizations for weak exponential stability properties of evolution operators in banach spaces variants for weak exponential stability of some well known results in uniform stability theory buse and niculescu 2009 1 daleckij and krein 1974 2 datko 1973 3 rolewicz 1986 7 stoica and megan 2009 8 are obtained

a discrete perron ta li type theorem for the dichotomy of evolution operators following the perron ta li line of results we give a characterization of the uniform exponential dichotomy property for the abstract continuous time evolution families using a discrete time approach

strictly ordered minimal subsets of a class of convex monotone skew product semiflows we study the topological and ergodic structure of a class of convex and monotone skew product semiflows we assume the existence of two strongly ordered minimal subsets k1 k2 and we obtain an ergodic representation of their upper lyapunov exponents in the case of null upper lyapunov exponents we obtain a lamination into minimal subsets of an intermediate region where the restriction of the semiflow is affine in the hyperbolic case we deduce the long time behaviour of every trajectory ordered with k2 some examples of skew product semiflows generated by non autonomous differential equations and satisfying the assumptions of monotonicity and convexity are also presented

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

evolutionary semigroups and dichotomy of linear skew product flows on locally compact spaces with banach fibers abstract we study evolutionary semigroups generated by a strongly continuous semi cocycle over a locally compact metric space acting on banach fibers this setting simultaneously covers evolutionary semigroups arising from non autonomous abstract cauchy problems and c 0 semigroups and linear skew product flows the spectral mapping theorem for these semigroups is proved the hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew product flow to this end a banach algebra of weighted composition operators is studied the results are applied in the study of roughness of the dichotomy dichotomy and solutions of nonhomogeneous equations green s function for a linear skew product flow pointwise dichotomy versus global dichotomy and evolutionary semigroups along trajectories of the flow

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations abstract we prove that the admissibility of any pair of vector valued schaffer function spaces satisfying a very general technical condition implies the existence of a no past exponential dichotomy for an exponentially bounded strongly continuous cocycle over a semiflow roughly speaking the class of schaffer function spaces consists in all function spaces which are invariant under the right shift and therefore our approach addresses most of the possible pairs of admissible spaces complete characterizations for the exponential dichotomy of cocycles are also obtained moreover we involve a concept of a no past exponential dichotomy for cocycles weaker than the classical concept defined by sacker and sell 1994 in 23 our definition of exponential dichotomy follows partially the definition given by chow and leiva 1996 in 4 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to o perron 1930 14 j daleckij and m krein 1974 7 j l massera and j j schaffer 1966 11 n van minh f rabiger and r schnaubelt 1998 26

a version of a theorem of r datko for nonuniform exponential contractions abstract we point out versions of a well known theorem of r datko for nonuniform exponential contractions more precisely we obtain theorems that characterize the nonuniform exponential stability and the nonuniform exponential dichotomy for nonuniform exponential contractions thus we generalize the results in datko 1972 8 and preda and megan 1985 14

uniqueness theorems of hermitian metrics of seminegative curvature on quotients of bounded symmetric domains in the theory of locally symmetric spaces of negative ricci curvature it has been a classical problem to study the extent to which the topology of the manifold determines the geometry and in the hermitian case the complex structure the rigidity theorem of mostow 26 asserts that for a compact locally symmetric riemannian manifold of negative ricci curvature the manifold is determined up to isometry and normalizing constants by its fundamental group among the class of such manifolds with the obvious exceptions involving compact riemann surfaces the same theorem for locally symmetric riemannian manifolds of finite volume and negative ricci curvature was proved by prasad 27 in case of rank 1 and included in the super rigidity theorem of margulis 13 in the case of rank 2 in the class of compact locally symmetric hermitian manifolds of negative ricci curvature calabi vesentini 5 and borel 3 proved vanishing theorems of certain cohomology groups which imply in particular that for x locally irreducible of complex dimension 2 there exists no non trivial deformation of x as a complex manifold in this direction siu 32 33 proved the strong rigidity of the kdhler manifold x in the sense that any compact kkhler manifold m homotopic to x is necessarily biholomorphic or conjugate biholomorphic to x if m is also locally symmetric by assumption then it follows from the uniqueness of kihler einstein metrics of negative ricci curvature yau 37 that the conjugate biholomorphism is in fact an isometry up to a normalizing constant it is thus natural to ask

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

on exponential h expansiveness of semigroups of operators in banach spaces this paper introduces the concept of exponential h expansiveness for semigroups of nonlinear operators which is an extension of classical concept of exponential expansiveness following the idea of obtaining an unitary treatment for stability and expansiveness necessary and sufficient conditions for exponential h expansiveness are given as particular cases the variants for exponential expansiveness of some well known stability results due to datko pazy ichikawa rolewicz and neerven are obtained

exponential dichotomy on the real line svd and qr methods abstract in this work we show when and how techniques based on the singular value decomposition svd and the qr decomposition of a fundamental matrix solution can be used to infer if a system enjoys or not exponential dichotomy on the whole real line

evolutionary semigroups and lyapunov theorems in banach spaces we present a spectral mapping theorem for continuous semigroups of operators on any banach space e the condition for the hyperbolicity of a semigroup on e is given in terms of the generator of an evolutionary semigroup acting in the space of e valued functions the evolutionary semigroup generated by the propagator of a nonautonomous differential equation in e is also studied a discrete technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity exponential dichotomy of the nonautonomous equation

on uniform exponential stability for skew evolution semiflows on banach spaces the aim of this paper is to give several characterizations for the property of stability for skew evolution semiflows on banach spaces thus we obtain generalizations of some well known results due to barbashin datko and rolewicz in the case of evolution equations in banach spaces a unified treatment in the uniform setting is provided

on holomorphic isometric embeddings of the unit n ball into products of two unit m balls we study holomorphic isometric embeddings of the complex unit n ball into products of two complex unit m balls with respect to their bergman metrics up to normalization constants the isometric constant there are two trivial holomorphic isometric embeddings for m n given by f1 z 0 in m z with the isometric constant equal to m 1 n 1 and f2 z in m z in m z with the isometric constant equal to 2 m 1 n 1 here i n m mathbb c n longrightarrow mathbb c m is the canonical embedding we prove that when m 2n these are the only holomorphic isometric embeddings up to unitary transformations