duality and twisted sums of banach spaces we give a negative answer to the three space problem for the banach space properties to be complemented in a dual space and to be isomorphic to a dual space solving a problem of vogt lectures held in the functional analysis seminar dusseldorf wuppertal jan feb 1987 and another posed by diaz et al in bull polish acad sci math 40 1992 221 224 precisely we construct an exact sequence 0 l2 d w 0 in which w is a separable dual and d is not isomorphic to a dual space we also show the existence of an exact sequence 0 y x z 0 where both y and z are dual spaces and x is not even complemented in its bidual to do that we perform a study of the basic questions on duality from the point of view of exact sequences of banach spaces

exact ground state energy of the strong coupling polaron the polaron has been of interest in condensed matter theory and field theory for about half a century especially the limit of large coupling constant a it was not until 1983 however that a proof of the asymptotic formula for the ground state energy was finally given by using difficult arguments involving the large deviation theory of path integrals here we derive the same asymptotic result e 0 c 2 and with explicit error bounds by simple rigorous methods applied directly to the hamiltonian our method is easily generalizable to other settings e g the excitonic and magnetic polarons

gibbs measures on brownian currents motivated by applications to quantum field theory we consider gibbs measures for which the reference measure is wiener measure and the interaction is given by a double stochastic integral and a pinning external potential in order properly to characterize these measures through dlr equations we are led to lift wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields we prove existence and basic properties of such gibbs measures in the small coupling regime by means of cluster expansion

an expression of the ground state energy of the spin boson model abstract an expression of the ground state energyesbof the spin boson hamiltonianhsbis considered the expression in the cases of both massive and massless bosons is given by a nonperturbative method by using the expression we show a necessary and sufficient condition with respect to a parameterg 1 0 such that a formula withgattains toesb

convergence of path measures arising from a mean field or polaron type interaction we discuss the limiting path measures of markov processes with either a mean field or a polaron type interaction of the paths in the polaron type situation the strength is decaying at large distances on the time axis and so the interaction is of short range in time in contrast in the mean field model the interaction is weak but of long range in time donsker and varadhan proved that for the partition functions there is a transition from the polaron type to the mean field interaction when passing to a limit by letting the strength tend to zero while increasing the range the discussion of the path measures is more subtle we treat the mean field case as an example of a differentiable interaction and discuss the transition from the polaron type to the mean field interaction for two instructive examples

equivalent classes of critical circles we present numerical evidence that the fractal properties of the critical invariant circles of a typical area preserving twist map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers and d q are numerically the same for all critical invariant circles of the standard and sine maps which have the rotation numbers with the same periodic tail

intermittency in a catalytic random medium in this paper we study intermittency for the parabolic anderson equation u t u u where u d 0 is the diffusion constant is the discrete laplacian and d 0 is a space time random medium we focus on the case where is times the random medium that is obtained by running independent simple random walks with diffusion constant starting from a poisson random field with intensity throughout the paper we assume that 0 the solution of the equation describes the evolution of a reactant u under the influence of a catalyst we consider the annealed lyapunov exponents that is the exponential growth rates of the successive moments of u and show that they display an interesting dependence on the dimension d and on the parameters with qualitatively different intermittency behavior in d 1 2 in d 3 and in d 4 special attention is given to the asymptotics of these lyapunov exponents for 0 and

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

uniformly exhaustive submeasures and nearly additive set functions every uniformly exhaustive submeasure is equivalent to a measure from this we deduce that every vector measure with compact range in an f space has a control measure we also show that co or any e space is a t space i e cannot be realized as the quotient of a nonlocally convex f space by a one dimensional subspace

stationary parabolic anderson model and intermittency this paper is devoted to the analysis of the large time behavior of the solutions of the anderson parabolic problem frac partial u partial t kappa delta u xi x u when the potential x is a homogeneous ergodic random field on d our goal is to prove the asymptotic spatial intermittency of the solution and for this reason we analyze the large time properties of all the moments of the positive solutions this provides an extension to the continuous space d of the work done originally by gartner and molchanov in the case of the lattice d in the process of our moment analysis we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized gaussian fields interpolating continuously between the exponent 2 which is found in the case of bona fide continuous gaussian fields x and the exponent 3 2 appearing in the case of a one dimensional white noise finally we also determine the precise almost sure large time asymptotics of the positive solutions

critical circle maps near bifurcation we estimate harmonic scalings in the parameter space of a one parameter family of critical circle maps these estimates lead to the conclusion that the hausdorff dimension of the complement of the frequency locking set is less than 1 but not less than 1 3 moreover the rotation number is a holder continuous function of the parameter

large deviations for noninteracting infinite particle systems a large deviation property is established for noninteracting infinite particle systems previous large deviation results obtained by the authors involved a singlei function because the cases treated always involved a unique invariant measure for the process in the context of this paper there is an infinite family of invariant measures and a corresponding infinite family ofi functions governing the large deviations

strong coupling limit for a certain class of polaron models it will be shown that a polaron like asymptotic is given for a class of functions which are called of decreasing interaction and nicely approximable with respect to the wiener process using large deviation techniques one can see that the asymptotics for thestrong coupling limit for those polaron models is given by a meanfield model and can be described by a variational problem

continuous integral kernels for unbounded schrodinger semigroups and their spectral projections abstract by suitably extending a feynman kac formula of simon canad math soc conf proc 28 2000 317 we study one parameter semigroups generated by the negative of rather general schrodinger operators which may be unbounded from below and include a magnetic vector potential in particular a common domain of essential self adjointness for such a semigroup is specified moreover each member of the semigroup is proven to be a maximal carleman operator with a continuous integral kernel given by a brownian bridge expectation the results are used to show that the spectral projections of the generating schrodinger operator also act as carleman operators with continuous integral kernels applications to schrodinger operators with rather general random scalar potentials include a rigorous justification of an integral kernel representation of their integrated density of states a relation frequently used in the physics literature on disordered solids

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

long time asymptotics of diffusion in random media and related problems for n dimensional diffusion in gaussian random fields with continuous and singular covariances the leading long time behaviour of the averaged population is derived by estimating brownian motion expectations it is shown how these results are related to the leading low energy behaviour of the density of states for a particle in a corresponding random potential and to the strong coupling limit of the free energy of the pekar frolich polaron

gibbs measures on mutually interacting brownian paths under singularities we are interested in the analysis of gibbs measures defined on two independent brownian paths in d interacting through a mutual self attraction this is expressed by the hamiltonian 2dv x y dx dy with two probability measures and representing the occupation measures of two independent brownian motions we will be interested in a class of potentials v that are singular e g dirac or coulomb type interactions in 3 or the correlation function of the parabolic anderson problem with white noise potential the mutual interaction of the brownian paths inspires a compactification of the quotient space of orbits of product measures which is structurally different from the self interacting case introduced in 27 owing to the lack of shift invariant structure in the mutual interaction we prove a strong large deviation principle for the product measures of two brownian occupation measures in such a compactification and derive asymptotic path behavior under gibbs measures on wiener paths arising from mutually attracting singular interactions for the spatially smoothened parabolic anderson model with white noise potential our analysis allows a direct computation of the annealed lyapunov exponents and a strict ordering of them implies the intermittency effect present in the smoothened model 2017 wiley periodicals inc

renormalization for golden circles we prove that twist maps of the cylinder that are attracted by any fixed point of mackay s renormalization operator have a transitive invariant golden circle provided the fixed point satisfies a few simple purely topological conditions these conditions can be verified by finite precision arithmetics they are fulfilled for the simple fixed point and seem to be fulfilled for the critical fixed point taking existence and hyperbolicity of the critical fixed point for granted we conclude that the standard map has a critical invariant golden circle the induced map on the circle is topologically conjugate to a rigid rotation we can show that the conjugator is holder continuous moreover it is not differentiable on a dense set of points

mean field interaction of brownian occupation measures ii a rigorous construction of the pekar process we consider mean field interactions corresponding to gibbs measures on interacting brownian paths in three dimensions the interaction is self attractive and is given by a singular coulomb potential the logarithmic asymptotics of the partition function for this model were identified in the 1980s by donsker and varadhan in terms of the pekar variational formula which coincides with the behavior of the partition function of the polaron problem under strong coupling based on this in 1986 spohn made a heuristic observation that the strong coupling behavior of the polaron path measure on certain time scales should resemble a process named as the pekar process whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean field measures under interest whose rigorous analysis remained open the present paper is devoted to a precise analysis of these mean field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the pekar variational problem this leads to a rigorous construction of the aforementioned pekar process and hence is a contribution to the understanding of the mean field approximation of the polaron problem on the level of path measures the method of our proof is based on the compact large deviation theory developed by mukherjee and varadhan in 2016 its extension to the uniform strong metric for the singular coulomb interaction was carried out by konig and mukherjee in 2015 as well as an idea inspired by a partial path exchange argument appearing in 1997 in work by bolthausen and schmock 2016 wiley periodicals inc

twisted sums with c k spaces if x is a separable banach space we consider the existence of non trivial twisted sums 0 c k y x 0 where k 0 1 or for the case k 0 1 we show that there exists a twisted sum whose quotient map is strictly singular if and only if x contains no copy of 1 if k we prove an analogue of a theorem of johnson and zippin for k 0 1 by showing that all such twisted sums are trivial if x is the dual of a space with summable szlenk index e g x could be tsirelson s space a converse is established under the assumption that x has an unconditional finite dimensional decomposition we also give conditions for the existence of a twisted sum with c with strictly singular quotient map

return time entropies for a class of circle homeomorphisms poincare recurrence for a class of circle maps is used to study the properties of the corresponding invariant measures in the subcritical case when the map is a diffeomorphism the return time measure is smooth and in the critical case when the map is only a homeomorphism the measure is only continuous furthermore in the considered class of critical maps the behaviour of the return time entropy depends only on the tail in the continued fraction expansion of the rotation number

singularity spectrum of a critical kam torus the authors numerically determine the f alpha spectrum of scaling indices for critical kam tori in the area preserving standard map similarities and differences between the corresponding orbits of circle maps are discussed

the dimension spectrum of some dynamical systems we analyze the dimension spectrum previously introduced and measured experimentally by jensen kadanoff and libchaber using large deviation theory we prove for some invariant measures of expanding markov maps that the hausdorff dimensionf of the set on which the measure has a singularity is a well defined concave and regular function in particular we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number we also show in these particular cases that the functionf is universal

stability of vector measures and twisted sums of banach spaces a banach space x is said to have the svm stability of vector measures property if there exists a constant v such that for any algebra of sets f and any function f x satisfying a b a b 1for disjoint a b f there is a vector measure f x with a a v for all a f if this condition is valid when restricted to set algebras f of cardinality less than some fixed cardinal number then we say that x has the svm property the least cardinal for which x does not have the svm property if it exists is called the svm character of x we apply the machinery of twisted sums and quasi linear maps to characterise these properties and to determine svm characters for many classical banach spaces we also discuss connections between the svm property injectivity and the three space problem

large deviations for stationary gaussian processes in their previous work on large deviations the authors always assumed the base process to be markovian whereas here they consider the base process to be stationary gaussian similar large deviation results are obtained under natural hypotheses on the spectral density function of the base process a rather explicit formula for the entropy involved is also obtained

singular measures in circle dynamics critical circle homeomorphisms have an invariant measure totally singular with respect to the lebesgue measure we prove that singularities of the invariant measure are of holder type the hausdorff dimension of the invariant measure is less than 1 but greater than 0

geometric methods for nonlinear many body quantum systems abstract geometric techniques have played an important role in the seventies for the study of the spectrum of many body schrodinger operators in this paper we provide a formalism which also allows to study nonlinear systems we start by defining a weak topology on many body states which appropriately describes the physical behavior of the system in the case of lack of compactness that is when some particles are lost at infinity we provide several important properties of this topology and use them to write a simple proof of the famous hvz theorem in the repulsive case in the second step we recall the method of geometric localization in fock space as proposed by derezinski and gerard and we relate this tool to our weak topology we then provide several applications we start by studying the so called finite rank approximation which consists in imposing that the many body wavefunction can be expanded using finitely many one body functions we thereby emphasize geometric properties of hartree fock states and prove nonlinear versions of the hvz theorem in the spirit of works of friesecke in the last section we study translation invariant many body systems comprising a nonlinear term which effectively describes the interactions with a second system as an example we prove the existence of the multi polaron in the pekar tomasevich approximation for certain values of the coupling constant

large deviations for independent random walks we consider a system of independent random walks on let n x be the number of particles atx at timen and letl n x 0 x n x be the total occupation time ofx by timen in this paper we study the large deviations ofl n 0 l n 1 the behavior we find is much different from that ofl n 0 we investigate the limiting behavior when the initial configurations has asymptotic density 1 and when 0 x are i i d poisson mean 1 finding that the asymptotics are different in these two cases

large deviations for the three dimensional super brownian motion let t dx denote a three dimensional super brownian motion with deterministic initial state 0 dx dx the lebesgue measure let v r 3 r be holder continuous with compact support not identically zero and such that r 3v x dx 0 we show that log p 0 t r 3v x s dx ds bt 3 4 is of order t 1 2 as t for b 0 this should be compared with the known result for the case r 3v x dx 0 in that case the normalization bt 3 4 b 0 must be replaced by bt b r 3v x dx in order that the same statement hold true while this result only captures the logarithmic order the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

colocality and twisted sums of banach spaces using the relation between subspaces of banach spaces and quotients of their duals we introduce the concept of colocality to give a new method that guarantees the existence of nontrivial twisted sums in which finite quotients play a major role theorem 1 7 an interesting point is that no restrictions are imposed on the quotients only on the various subspaces new examples of nontrivial twisted sums are given

monotonicity of the polaron energy ii general theory of operator monotonicity we construct a general theory of operator monotonicity and apply it to the frohlich polaron hamiltonian this general theory provides a consistent viewpoint of the frohlich model

large deviations for trapped interacting brownian particles and paths are defined in terms of transformed path measures on finite time intervals under a trap hamiltonian and two respective pair interaction hamiltonians the first pair interaction exhibits a particle repellency while the second one imposes a path repellency we analyze both models in the limit of diverging time with fixed number n of brownian motions in particular we prove large deviations principles for the normalized occupation measures the minimizers of the rate functions are related to a certain associated operator the hamilton operator for a system of n interacting trapped particles more precisely in the particle repellency model the minimizer is its ground state and in the pathrepellency model the minimizers are its ground product states in the case of path repellency we also discuss the case of a dirac type interaction which is rigorously defined in terms of brownian intersection local times we prove a large deviation result for a discrete variant of the model this study is a contribution to the search for a mathematical formulation of the quantum system of n trapped interacting bosons as a model for bose einstein condensation motivated by the success of the famous 1995 experiments recently lieb et al described the large n behavior of the ground state in terms of the well known gross pitaevskii formula involving the scattering length of the pair potential we prove that the large n behavior of the ground product states is also described by the gross pitaevskii formula however with the scattering length of the pair potential replaced by its integral

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

universal scaling of critical quasiperiodic orbits in a class of twist maps recently we have shown that the fractal properties of the critical invariant circles of the standard map as summarized by the spectrum and the generalized dimensions d q depend only on the tails in the continued fraction expansion of the corresponding rotation numbers in buric n mudrinic m and todorovic k 1997 30 l161 in the present paper this result is extended on the whole class of sufficiently smooth area preserving twist maps of cylinders we present numerical evidence that the and d q are the same for all critical invariant circles of any such map which have the rotation numbers with the same tail