finite rank bratteli diagrams structure of invariant measures we consider bratteli diagrams of finite rank not necessarily simple and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces it is shown that every ergodic invariant measure finite or regular infinite is obtained by an extension from a simple subdiagram we further investigate quantitative properties of these measures which are mainly determined by the asymptotic behavior of products of incidence matrices a number of sufficient conditions for unique ergodicity are obtained one of these is a condition of exact finite rank which parallels a similar notion in measurable dynamics several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them we then prove that the vershik map on the path space of an exact finite rank diagram cannot be strongly mixing independent of the ordering on the other hand for the so called consecutive ordering the vershik map is not strongly mixing on all finite rank diagrams msc 37b05 37a25 37a20

representations of higher rank graph c algebras associated to λ semibranching function systems abstract in this paper we discuss a method of constructing separable representations of the c algebras associated to strongly connected row finite k graphs we begin by giving an alternative characterization of the semibranching function systems introduced in an earlier paper with an eye towards constructing such representations that are faithful our new characterization allows us to more easily check that examples satisfy certain necessary and sufficient conditions we present a variety of new examples relying on this characterization we then use some of these methods and a direct limit procedure to identify a faithful separable representation for any row finite source free k graph

derivation of hartree s theory for generic mean field bose systems in this paper we provide a novel strategy to prove the validity of hartree s theory for the ground state energy of bosonic quantum systems in the mean field regime for the well known case of trapped bose gases this can be shown using the strong quantum de finetti theorem which gives the structure of infinite hierarchies of k particles density matrices here we deal with the case where some particles are allowed to escape to infinity leading to a lack of compactness our approach is based on two ingredients 1 a weak version of the quantum de finetti theorem and 2 geometric techniques for many body systems our strategy does not rely on any special property of the interaction between the particles in particular our results cover those of benguria lieb and lieb yau for respectively bosonic atoms and boson stars

separable representations kms states and wavelets for higher rank graphs letbe a strongly connected finite higher rank graph in this paper we construct faithful representations of c on certain separable hilbert spaces of the form l 2 x by introducing the notion of a semibranching function system a generalization of the semibranching function systems studied by marcolli and paolucci in particular we obtain a faithful representation of c on l 2 1 m where m is the perron frobenius probability measure on the infinite path space 1 recently studied by an huef laca raeburn and sims we also show how a semibranching function system gives rise to kms states for c for the higher rank graphs of robertson and steger we also obtain a faithful representation of c on l 2 x where x is a fractal subspace of 0 1 by embedding 1 into 0 1 as a fractal subset x of 0 1 in this latter case we additionally show that there exists a kms state for c whose inverse temperature is equal to the hausdorff dimension of x finally we construct a wavelet system for l 2 1 m by generalizing the work of marcolli and paolucci from graphs to higher rank graphs

the perron frobenius operators invariant measures and representations of the cuntz krieger algebras for a transformation f on a measure space x we show that the perron frobenius operator of f can be written by a representation l2 x of the cuntz krieger algebra oa associated with f when f satisfies some assumption especially when oa is the cuntz algebra on and l2 x in the above is some irreducible representation of on then there is an f invariant measure on x which is absolutely continuous with respect to

on time dependent schrodinger equations global well posedness and growth of sobolev norms in this paper we consider time dependent schrodinger linear pdes of the form i t l t where l t is a continuous family of self adjoint operators we give conditions for well posedness and polynomial growth for the evolution in abstract sobolev spaces if l t h v t where v t is a perturbation smooth in time and h is a self adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing we prove that the sobolev norms of the solution grow at most as t when t for any 0 if v t is analytic in time we improve the bound to log t for some 0 the proof follows the strategy due to howland joye and nenciu of the adiabatic approximation of the flow we recover most of known results and obtain new estimates for several models including 1 degree of freedom schrodinger operators on r and schrodinger operators on zoll manifolds

non uniform continuity of the solution map to the rotation two component camassa holm system abstract studied in this paper is the rotation two component camassa holm system which can be considered as a model for the propagation of long crested shallow water waves in the equatorial ocean regions with the weak coriolis effect due to the earth s rotation it is shown that the data to solution map is not uniformly continuous on both the line and the circle in certain sobolev spaces depending on the effect of the coriolis force and stretching balance parameters the non uniform dependence result is established by using the method of approximate solutions with refined energy estimates

quantum fluctuations and rate of convergence towards mean field dynamics the nonlinear hartree equation describes the macroscopic dynamics of initially factorized n boson states in the limit of large n in this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting hartree dynamics more precisely we prove bounds on the difference between the one particle density associated with the solution of the n body schrodinger equation and the orthogonal projection onto the solution of the hartree equation

a central limit theorem in many body quantum dynamics we study the many body quantum evolution of bosonic systems in the mean field limit the dynamics is known to be well approximated by the hartree equation so far the available results have the form of a law of large numbers in this paper we go one step further and we show that the fluctuations around the hartree evolution satisfy a central limit theorem interestingly the variance of the limiting gaussian distribution is determined by a time dependent bogoliubov transformation describing the dynamics of initial coherent states in a fock space representation of the system

von neumann algebras of strongly connected higher rank graphs we investigate the factor types of the extremal kms states for the preferred dynamics on the toeplitz algebra and the cuntz krieger algebra of a strongly connected finite k graph for inverse temperatures above 1 all of the extremal kms states are of type mathrm i infty at inverse temperature 1 there is a dichotomy if the k graph is a simple k dimensional cycle we obtain a finite type mathrm i factor otherwise we obtain a type iii factor whose connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph

quantitative derivation of the gross pitaevskii equation starting from first principle many body quantum dynamics we show that the dynamics of bose einstein condensates can be approximated by the time dependent nonlinear gross pitaevskii equation giving a bound on the rate of the convergence initial data are constructed on the bosonic fock space applying an appropriate bogoliubov transformation on a coherent state with expected number of particles mathit n the bogoliubov transformation plays a crucial role it produces the correct microscopic correlations among the particles our analysis shows that on the level of the one particle reduced density the form of the initial data is preserved by the many body evolution up to a small error that vanishes as mathit n 1 2 in the limit of large mathit n

uct kirchberg algebras have nuclear dimension one we prove that every kirchberg algebra in the uct class has nuclear dimension 1 we first show that kirchberg 2 graph algebras with trivial k0 and finite k1 have nuclear dimension 1 by adapting a technique developed by winter and zacharias for cuntz algebras we then prove that every kirchberg algebra in the uct class is a direct limit of 2 graph algebras to obtain our main theorem

kms states on the c algebra of a higher rank graph and periodicity in the path space we study the kms states of the c algebra of a strongly connected finite k graph we find that there is only one 1 parameter subgroup of the gauge action that can admit a kms state the extreme kms states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite path space of the graph we deduce that there is a unique kms state if and only if the k graph c algebra is simple giving a complete answer to a question of yang when the k graph c algebra is not simple our results reveal a phase change of an unexpected nature in its toeplitz extension

mean field dynamics of a quantum tracer particle interacting with a boson gas abstract we consider the dynamics of a heavy quantum tracer particle coupled to a non relativistic boson field in r 3 the pair interactions of the bosons are of mean field type with coupling strength proportional to 1 n where n is the expected particle number assuming that the mass of the tracer particle is proportional to n we derive generalized hartree equations in the limit n moreover we prove the global well posedness of the associated cauchy problem for sufficiently weak interaction potentials

quasidiagonality of nuclear c algebras we prove that faithful traces on separable and nuclear c algebras in the uct class are quasidiagonal this has a number of consequences firstly by results of many hands the classification of unital separable simple and nuclear c algebras of finite nuclear dimension which satisfy the uct is now complete secondly our result links the finite to the general version of the toms winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension finally we confirm the rosenberg conjecture discrete amenable groups have quasidiagonal c algebras

mean field dynamics singular potentials and rate of convergence we consider the time evolution of a system of n identical bosons whose interaction potential is rescaled by n 1 we choose the initial wave function to describe a condensate in which all particles are in the same one particle state it is well known that in the mean field limit n the quantum n body dynamics is governed by the nonlinear hartree equation using a nonperturbative method we extend previous results on the mean field limit in two directions first we allow a large class of singular interaction potentials as well as strong possibly time dependent external potentials second we derive bounds on the rate of convergence of the quantum n body dynamics to the hartree dynamics

an analogue of cobham s theorem for graph directed iterated function systems abstract feng and wang showed that two homogeneous iterated function systems in r with multiplicatively independent contraction ratios necessarily have different attractors in this paper we extend this result to graph directed iterated function systems in r n with contraction ratios that are of the form 1 for integers by using a result of boigelot et al this allows us to give a proof of a conjecture of adamczewski and bell in doing so we link the graph directed iterated function systems to buchi automata in particular this link extends to real numbers we introduce a logical formalism that permits to characterize sets of r n whose representations in base are recognized by some buchi automata this result depends on the algebraic properties of the base being a pisot or a parry number the main motivation of this work is to draw a general picture representing the different frameworks where an analogue of cobham s theorem is known

fluctuations around hartree states in the mean field regime we consider the dynamics of a large system of n interacting bosons in the mean field regime where the interaction is of order 1 n we prove that the fluctuations around the nonlinear hartree state are generated by an effective quadratic hamiltonian in fock space which is derived from bogoliubov s approximation we use a direct method in the n particle space which is different from the one based on coherent states in fock space

norm approximation for many body quantum dynamics focusing case in low dimensions abstract we study the norm approximation to the schrodinger dynamics of n bosons in r d d 1 2 with an interaction potential of the form n d 1 w n x y here we are interested in the focusing case w 0 assuming that there is complete bose einstein condensation in the initial state we show that in the large n limit the evolution of the condensate is effectively described by a nonlinear schrodinger equation and the evolution of the fluctuations around the condensate is governed by a quadratic hamiltonian resulting from bogoliubov approximation our result holds true for all 0 when d 1 and for all 0 1 when d 2

monic representations of the cuntz algebra and markov measures abstract we study representations of the cuntz algebras o n while for fixed n the set of equivalence classes of representations of o n is known not to have a borel cross section there are various subclasses of representations which can be classified we study monic representations of o n that have a cyclic vector for the canonical abelian subalgebra we show that o n has a certain universal representation which contains all positive monic representations a large class of examples of monic representations is based on markov measures we classify them and as a consequence we obtain that different parameters yield mutually singular markov measure extending the classical result of kakutani the monic representations based on the kakutani measures are exactly the ones that have a one dimensional cyclic s i invariant space

graph c algebras branching systems and the perron frobenius operator in this paper we show how to produce a large number of representations of a graph c algebra in the space of the bounded linear operators in l2 x these representations are very concrete and in the case of graphs that satisfy condition l we use our techniques to realize the associated graph c algebra as a subalgebra of the bounded operators in l2 r we also show how to describe the perron frobenius operator of ergodic theory in terms of the representations we associate to a graph