kms states of quasi free dynamics on pimsner algebras abstract a continuous one parameter group of unitary isometries of a right hilbert c bimodule induces a quasi free dynamics on the cuntz pimsner c algebra of the bimodule and on its toeplitz extension the restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial and the corresponding kms states of the toeplitz cuntz pimsner and cuntz pimsner c algebras are characterized in terms of traces on the algebra of coefficients this generalizes and sheds light onto various earlier results about kms states of the gauge actions on cuntz algebras cuntz krieger algebras and crossed products by endomorphisms we also obtain a more general characterization in terms of kms weights for the case in which the inducing isometries are not unitary and accordingly the restriction of the quasi free dynamics to the algebra of coefficients is nontrivial

partial actions and kms states on relative graph c algebras abstract the relative graph c algebras introduced by muhly and tomforde are generalizations of both graph algebras and their toeplitz extensions for an arbitrary graph e and a subset r of the set of regular vertices of e we show that the relative graph c algebra c e r is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space given a time evolution on c e r induced by a function on the edge set we characterize the kms states and ground states using an abstract result of exel and laca guided by their work on kms states for toeplitz cuntz krieger type algebras associated to infinite matrices we obtain complete descriptions of the convex sets of kms states of finite type and of kms states of infinite type whose associated measures are supported on recurrent infinite paths this allows us to give a complete concrete description of the convex set of all kms states for a big class of graphs which includes all graphs with finitely many vertices

actions of symbolic dynamical systems on c algebras ii simplicity of c symbolic crossed products and some examples a c symbolic dynamical system mathcal a rho sigma consists of a unital c algebra mathcal a and a finite family rho alpha alpha in sigma of endomorphisms of mathcal a indexed by symbols of satisfying some conditions the endomorphisms rho alpha alpha in sigma yield both a subshift and a c algebra of a hilbert c bimodule the obtained c algebra is regarded as a crossed product of mathcal a by the subshift we will study simplicity condition of these c algebras some examples such as irrational rotation cuntz krieger algebras will be studied

equilibrium states and entropy theory for nica pimsner algebras abstract we study the equilibrium simplex of nica pimsner algebras arising from product systems of finite rank on the free abelian semigroup first we show that every equilibrium state has a convex decomposition into parts parametrized by ideals on the unit hypercube secondly we associate every gauge invariant part to a sub simplex of tracial states of the diagonal algebra we show how this parametrization lifts to the full equilibrium simplices of non infinite type the finite rank entails an entropy theory for identifying the two critical inverse temperatures a the least upper bound for existence of non finite type equilibrium states and b the least positive inverse temperature below which there are no equilibrium states at all we show that the first one can be at most the strong entropy of the product system whereas the second is the infimum of the tracial entropies modulo negative values thus phase transitions can happen only in between these two critical points and possibly at zero temperature

fundamental group of simple c algebras with unique trace abstract we introduce the fundamental group f a of a unital simple c algebra a with a unique normalized trace we compute fundamental groups f a of several nuclear or non nuclear c algebras a k theoretical obstruction enables us to compute the fundamental group easily our study is essentially based on the computation of picard groups by kodaka

chaotic dynamics of continuous time topological semi flows on polish spaces abstract differently from lyapunov exponents li yorke devaney and others that appeared in the literature we introduce the concept chaos for a continuous semi flow f r x x on a polish space x with a metric d which is useful in the theory of ode and is invariant under topological equivalence of semi flows our definition is weaker than devaney s one since here f may have neither fixed nor periodic elements but it implies repeatedly observable sensitive dependence on initial data there is an 0 such that for any x x there corresponds a dense g set s u x in x satisfying lim sup t d f t x f t y y s u x this sensitivity is obviously stronger than guckenheimer s one that requires only d f t x f t y for some moment t 0 and some y arbitrarily close to x

kms states on c algebras associated to a family of commuting local homeomorphisms abstract we consider a family of commuting local homeomorphisms on a compact space and build a compactly aligned product system of hilbert bimodules the nica toeplitz algebra of this system carries a gauge action of a higher dimensional torus and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus we study the kms states of these dynamics for large inverse temperatures including we describe the simplex of kms states on the nica toeplitz algebra we illustrate our main theorem by considering backward shifts on the infinite path spaces of a class of k graphs whose shift maps commute

on the c algebra of a locally injective surjection and its kms states it is shown that a locally injective surjection on a compact metric space is a factor of a local homeomorphism in such a way that the associated c algebras are isomorphic this is subsequently used to obtain upper and lower bounds for the possible values of kms states for generalized gauge actions on the c algebra

monte carlo simulations of the ising spin glass on lattices with finite connectivity the authors have simulated the ising spin glass model on a random lattice with a finite average coordination number and also on the bethe lattice with various different boundary conditions in particular they have calculated the overlap function p q for two independent samples for the random lattice the results are consistent with a spin glass transition above which p q converges to a dirac delta function for large n number of sites and below which p q has in addition a long tail similar to previous results obtained for the infinite range model for the bethe lattice they obtain results in the interior by discarding the two outer shells of the cayley tree when calculating the thermal averages for fixed uncorrelated boundary conditions p q seems to converge to a delta function even below the spin glass transition whereas on a closed lattice correlated boundary conditions p q has a long tail similar to its behaviour in the random lattice case

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

devaney chaos li yorke chaos and multi dimensional li yorke chaos for topological dynamics abstract let t x x written t x be a topological semiflow flow on a uniform space x with t a multiplicative topological semigroup group not necessarily discrete we then prove if t x is non minimal topologically transitive with dense almost periodic points then it is sensitive to initial conditions as a result of this devaney chaos sensitivity to initial conditions for this very general setting let r x be a c 0 semiflow on a polish space then we show if r x is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior then it is multi dimensional li yorke chaotic that is there is a uncountable set x such that for any k 2 and any distinct points x 1 x k one can find two time sequences s n t n with s n x 1 x k x 1 x k x k and t n x 1 x k p p x k consequently devaney chaos multi dimensional li yorke chaos moreover let x be a non singleton polish space then we prove any weakly mixing c 0 semiflow r x is densely multi dimensional li yorke chaotic any minimal weakly mixing topological flow t x with t abelian is densely multi dimensional li yorke chaotic any weakly mixing topological flow t x is densely li yorke chaotic we in addition construct a completely li yorke chaotic minimal sl 2 r acting flow on the compact metric space r our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup group t

noncommutative pressure and the variational principle in cuntz krieger type c algebras let a be a self adjoint element of an exact c algebra a and a a a contractive completely positive map we define a notion of dynamical pressure p a which adopts voiculescu s approximation approach to noncommutative entropy and extends the voiculescu brown topological entropy and neshveyev and stormer unital nuclear pressure a variational inequality bounding p a below by the free energies h a with respect to the sauvageot thouvenot entropy h is established in two stages via the introduction of a local state approximation entropy whose associated free energies function as an intermediate term pimsner c algebras furnish a framework for investigating the variational principle which asserts the equality of p a with the supremum of the free energies over all invariant states in one direction we extend brown s result on the constancy of the voiculescu brown entropy upon passing to the crossed product and in another we show that the pressure of a self adjoint element over the markov subshift underlying the canonical map on the cuntz krieger algebra oa is equal to its classical pressure the latter result is extended to a more general setting comprising an expanded class of cuntz krieger type pimsner algebras leading to the variational principle for self adjoint elements in a diagonal subalgebra equilibrium states are constructed from kms states under certain conditions in the case of cuntz krieger algebras

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

a mean field spin glass with short range interactions we formulate and study a spin glass model on the bethe lattice appropriate boundary fields replace the traditional self consistent methods they give our model well defined thermodynamic properties we establish that there is a spin glass transition temperature above which the single site magnetizations vanish and below which the edwards anderson order parameter is strictly positive in a neighborhood below the transition temperature we use bifurcation theory to establish the existence of a nontrivial distribution of single site magnetizations two properties of this distribution are studied the leading perturbative correction to the gaussian scaling form at the transition and the nonperturbative behavior of the tails

partial dynamical systems and the kms condition given a countably infinite 0 1 matrix a without identically zero rows let a be the cuntz krieger algebra recently introduced by the authors and a be the toeplitz extension of a once the latter is seen as a cuntz pimsner algebra as recently shown by szymanski we study the kms equilibrium states of c dynamical systems based on a and a with dynamics satisfying for the canonical generating partial isometries s x and arbitrary real numbers n x 1 the kms states on both a and a are completely characterized for certain values of the inverse temperature according to the position of relative to three critical values defined to be the abscissa of convergence of certain dirichlet series associated to a and the n x our results for a are derived from those for a by virtue of the former being a covariant quotient of the latter when the matrix a is finite these results give theorems of olesen and pedersen for n and of enomoto fujii and watatani for a as particular cases

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

kms states on nica toeplitz hbox c c algebras given a quasi lattice ordered group g p and a compactly aligned product system x of essential hbox c correspondences over the monoid p we show that there is a bijection between the gauge invariant hbox kms beta states on the nica toeplitz algebra mathcal nt x of x with respect to a gauge type dynamics on one side and the tracial states on the coefficient algebra a satisfying a system in general infinite of inequalities on the other this strengthens and generalizes a number of results in the literature in several directions we do not make any extra assumptions on p and x and our result can in principle be used to study kms states at any finite inverse temperature beta under fairly general additional assumptions we show that there is a critical inverse temperature beta c such that for beta beta c all hbox kms beta states are of gibbs type hence gauge invariant in which case we have a complete classification of hbox kms beta states in terms of tracial states on a while at beta beta c we have a phase transition manifesting itself in the appearance of hbox kms beta states that are not of gibbs type in the case of right angled artin monoids we show also that our system of inequalities for traces on a can be reduced to a much smaller system a finite one when the monoid is finitely generated most of our results generalize to arbitrary quasi free dynamics on mathcal nt x

kms states entropy and the variational principle in full c dynamical systems to any periodic and full c dynamical system an invertible operator s acting on the banach space of trace functionals of the fixed point algebra is canonically associated kms states correspond to positive eigenvectors of s a perron frobenius type theorem asserts the existence of kms states at inverse temperatures equals the logarithms of the inner and outer spectral radii of s extremal kms states examples arising from subshifts in symbolic dynamics self similar sets in fractal geometry and noncommutative metric spaces are discussed

classification of certain infinite simple c algebras the class of c algebras that arise as the crossed product of a stable simple af algebra with a z action determined by an automorphism which maps a projection in the algebra onto a proper subprojection is proved to consist of simple purely infinite c algebras and a specific subclass of it is proved to be classified by k theory this subclass is large enough to exhaust all possible k groups if g0 and g1 are countable abelian groups with g1 torsion free as it must be then there is a c algebras a in the classified subclass with k0 a g0 and k 1 a g1 the subclass contains the cuntz algebras o with n even and the cuntz krieger algebras o a with k0 o a of finite old order and it is closed under forming inductive limits the c algebras in the classified subclass can be viewed as classifiable models in a strong sense of general simple purely infinite c algebras with the same k theory

chaotic oscillations of a linear hyperbolic pde with a general nonlinear boundary condition abstract this article establishes a theorem that guarantees the occurrence of chaotic oscillations in a system governed by a linear hyperbolic partial differential equation pde with a nonlinear boundary condition nbc compared with the nbcs in all previous related literature such an nbc is more general both the left end and the right end of the system parameter interval for the occurrence of chaotic oscillations are precisely characterized the chaotic results obtained are further applied to two specific nbcs and the telegraph equation finally numerical examples verify the effectiveness of theoretical prediction

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

on chaotic minimal center of attraction of a lagrange stable motion for topological semi flows abstract let f r x x be a topological semi flow on a polish space x in 1977 karl sigmund conjectured that if there is a point x in x such that the motion f x has just x as its minimal center of attraction then all such x form a residual subset of x in this paper we first present a positive solution to this conjecture and then apply it to the study of chaotic dynamics occurring inside or near the minimal center of attraction of a motion f x

nonisotropic chaotic oscillations of the wave equation due to the interaction of mixing transport term and superlinear boundary condition abstract this paper studies the nonisotropic chaotic oscillations of the initial boundary value problem of one dimensional wave equation with a mixing transport term it separately considers that the boundary condition at the right end of the wave equation is a superlinear type and linear perturbation of such type each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term for each type of boundary condition the occurrence of nonisotropic chaotic oscillations is rigorously proved numerical examples verify the effectiveness of theoretical prediction

application of statistical mechanics to combinatorial optimization problems the chromatic number problem and q partitioning of a graph methods of statistical mechanics are applied to two important np complete combinatorial optimization problems the first is the chromatic number problem which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color the second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections both models are mapped into a frustrated potts model which is related to theq state potts spin glass for the first problem we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation the quenched model is also discussed for the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq 3 and 4 potts spin glass using parisi s replica symmetry breaking we also perform some numerical simulations to test the theoretical result and obtain very good agreement

topological entropy and the af core of a graph c algebra abstract let c e be the c algebra associated with a locally finite directed graph e and a e be the af core of c e for the topological entropy ht e in the sense of brown voiculescu of the canonical completely positive map e on the graph c algebra it is known that if e is finite ht e ht e a e h b e h l e where h b e respectively h l e is the block respectively the loop entropy of e in case e is irreducible and infinite h l e ht e a e h b e t is known recently where e t is the graph e with the edges directed reversely then by monotonicity of entropy h l e ht e is clear in this paper we show that ht e h b e t holds for locally finite infinite graphs e the af core a e is known to be stably isomorphic to the graph c algebra c e c z of certain skew product e c z and we also show that ht e c z ht e a e examples e p p 1 of irreducible graphs with ht e p log p are discussed

replica symmetry breaking in finite connectivity systems a large connectivity expansion at finite and zero temperature parisi replica symmetry breaking is extended to random bond systems with finite fixed connectivity m 1 the free energy f is explicitly calculated for large m within the first stage of symmetry breaking at finite temperature t the 1 m expansion is found to diverge as t to o indeed direct evaluation at zero temperature shows that the large m expansion is in powers of 1 m introduction of symmetry breaking brings the value of f close approximately 1 for 10 m 20 to numerical estimates of banavar et al the same techniques apply to systems with average finite connectivity

the flow of weights and the cuntz pimsner algebras we describe the flows of weights for von neumann algebras arising from kms states for the gauge actions of the cuntz pimsner algebras in terms of poisson tail boundaries

on the existence of conformal measures a general notion of conformal measure is introduced and some basic properties are studied sufficient conditions for the existence of these measures are obtained using a general construction principle the geometric properties of conformal measures relate equilibrium states and hausdorff measures this is shown for invariant subsets of s under expanding maps

q partitioning of graphs with finite coordination number the np complete optimisation problem of q partitioning of graphs with finite connectivity is discussed using the effective local field method the authors obtain the local field distributions with and without the continuous part and use them to calculate the ground state energies of the three state potts spin glass on lattices with finite coordination number equal to three this in turn gives analytic results for the optimal cost of the 3 partitioning of graphs they perform simulations of actual 3 partitions of random graphs to compare with their theoretical results and obtain very good agreement ways for further improvement of the estimates are discussed

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

existence of chaotic oscillations in second order linear hyperbolic pdes with implicit boundary conditions abstract this paper establishes rigorously mathematical theorems that guarantee the existence of chaotic oscillations in the systems of second order linear hyperbolic pdes it separately considers the systems with nonlinear explicit boundary conditions ebcs and nonlinear implicit boundary conditions ibcs as well as those with such ibcs subjected to small perturbations where ibcs include ebcs as special cases but the latter cannot in general be expressed by the former numerical examples are demonstrated to illustrate the effectiveness of theoretical results

chaotic dynamics of minimal center of attraction of discrete amenable group actions abstract let g be a discrete infinite amenable group which acts from the left on a compact metric space x in this paper we study the chaotic dynamics exhibited inside and near a minimal center of attraction of g x relative to any folner net in g