uniform compact attractors for the coupled suspension bridge equations in this paper we investigate the uniformly absorbing set for the coupled suspension bridge equations with a new class of external forces and then prove the existence of the uniform attractors for the family of process corresponding to the equations

long time behavior of a suspension bridge equations with past history in this paper we study a suspension bridge equation with memory effects for the suspension bridge equation without memory there are many classical results existing results mainly devoted to existence and uniqueness of a weak solution energy decay of solution and existence of global attractors however the existence of global attractors for the suspension bridge equation with memory was no yet considered the object of the present paper is to provide some results on the well posedness and long time behavior to the suspension bridge equation when the unique damping mechanism is given by the memory term

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

entropy and the variational principle for actions of sofic groups recently lewis bowen introduced a notion of entropy for measure preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy by applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants and we establish the variational principle in this context in the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invertible in the full group c algebra

fuglede kadison determinants and entropy for actions of discrete amenable groups consider a discrete group t and an element in the integral group ring zt then t acts from the left on the discrete additive group zt ztf by automorphisms of groups dualizing we obtain a left t action on the compact pontrjagin dual group xf zt ztf by continuous automorphisms of groups by definition xf is a closed subshift of s1 1 for measurable or topological actions of amenable groups a good entropy theory is available ow m thus from now on let t be a finitely generated discrete amenable group we are interested in determining the topological entropy hf of the t action on xf it agrees with the measure theoretic entropy of the t action on xf equipped with its haar probability measure in their paper fk from 1952 fuglede and kadison introduced a determinant for units in finite factor von neumann algebras more recently l ck has general ized their theory to nonunits and to group von neumann algebras which are not necessarily factors l2 he needed this to define l2 torsion for general covering spaces the von neumann algebra a t of the amenable group t is a certain completion of cr and hence we may consider the fuglede kadison l ck determinant detjvr of we refer to section 6 for the definition of a log strong f0lner sequence it is known that finitely generated virtually nilpotent groups have log strong f0lner sequences our main result is

existence of strong solutions and global attractors for the coupled suspension bridge equations abstract in this paper we show the existence of the strong solutions for the coupled suspension bridge equations furthermore existence of the strong global attractors is investigated using a new semigroup scheme since the solutions of the coupled equation have no higher regularity and the semigroup associated with the solutions is not continuous in the strong hilbert space the results are new and appear to be optimal

mathematical analysis of dynamic models of suspension bridges in this paper we present some mathematical analysis of dynamic pde models of suspension bridges as proposed by lazer and mckenna our results are illustrated by numerical simulation with physical interpretation

pullback attractors for a non autonomous incompressible non newtonian fluid this paper studies the pullback asymptotic behavior of solutions for a non autonomous incompressible non newtonian fluid in two dimensional 2d bounded domains we first prove the existence of pullback attractors av in space v has h2 regularity see notation in section 2 and ah in space h has l2 regularity for the cocycle corresponding to the solutions of the fluid then we verify the regularity of the pullback attractors by showing av ah which implies the pullback asymptotic smoothing effect of the fluid in the sense that the solutions become eventually more regular than the initial data

local characteristics entropy and limit theorems for spanning trees and domino tilings via transfer impedances let g be a finite graph or an infinite graph on which mathbb z d acts with finite fundamental domain if g is finite let mathbf t be a random spanning tree chosen uniformly from all spanning trees of g if g is infinite methods from pemantle show that this still makes sense producing a random essential spanning forest of g a method for calculating local characteristics i e finite dimensional marginals of mathbf t from the transfer impedance matrix is presented this differs from the classical matrix tree theorem in that only small pieces of the matrix n dimensional minors are needed to compute small n dimensional marginals calculation of the matrix entries relies on the calculation of the green s function for g which is not a local calculation however it is shown how the calculation of the green s function may be reduced to a finite computation in the case when g is an infinite graph admitting a z d action with finite quotient the same computation also gives the entropy of the law of mathbf t these results are applied to the problem of tiling certain lattices by dominos the so called dimer problem another application of these results is to prove modified versions of conjectures of aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity included is a generalization of moments to tree valued random variables and criteria for these generalized moments to determine a distribution

a huygens principle for diffusion and anomalous diffusion in spatially extended systems we present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media for anisotropic systems strong chaos leads to diffusive behavior brownian motion with drift and weak chaos leads to superdiffusive behavior levy processes with drift for isotropic systems the drift term vanishes and strong chaos again leads to brownian motion we establish the existence of a nonlinear huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions

pullback d attractors for non autonomous suspension bridge equations abstract in this paper we prove the existence of a pullback d attractor in v h for the non autonomous suspension bridge equations

choosing a spanning tree for the integer lattice uniformly consider the nearest neighbor graph for the integer lattice zd in d dimensions for a large finite piece of it consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees as the piece gets larger this approaches a limiting measure on the set of spanning graphs for zd this is shown to be a tree if and only if d 4 in this case the tree has only one topological end that is there are no doubly infinite paths when d 2 5 the spanning forest has infinitely many components almost surely with each component having one or two topological ends

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

homoclinic points of algebraic ℤ actions an algebraic zd action is an action of zd by continuous automorphisms of a compact abelian group the dynamics of a single group automorphism have been investigated in great detail over the past several decades more recently the study of algebraic zd actions for d 2 has revealed a striking interplay between these actions and commutative algebra in 2 we summarize those parts of this interaction needed here the purpose of this paper is to study the homoclinic points of algebraic zd actions let al be an algebraic zd action on the compact abelian group x and let ox denote the additive identity of x a point x e x is homoclinic for ce if c nx ox as llnll oo the set ai x of all homoclinic points for oc is clearly a subgroup of x which we call the homoclinic group of oz in 3 we discuss some elementary properties of the homoclinic group including counltability of za x whenever oz is expansive our two main results are contained in 4 these are that if oz is an expansive algebraic ed action then 1 a x is nontrivial if and only if oz has strictly positive entropy and 2 a x is nontrivial and dense in x if and only if oz has completely positive entropy the second result is proved by first establishing in lemma 4 5 the density of a x for certain principal expansive actions by use of fourier analysis for these actions a x is generated by a single fundamental homoclinic point which can be computed explicitly this lemma is then combined with some commutative algebra to prove 2 recent work of kaminker and putnam 3 11 has suggested a general duality in the k theory of c algebras for a principal expansive action al on x we show that a x is isomorphic to the dual group of x providing a class of examples to which their duality theory applies ruelle has investigated expansive topological zd actions which satisfy an orbit tracing property called specification see 12 and 13 showing that there is a thermodynamic formalism for such actions in 5 we show that expansive algebraic zd actions with completely positive entropy always satisfy very strong specification properties thereby providing an extensive class of examples to which the thermodynamic formalism applies

harmonic models and spanning forests of residually finite groups abstract we prove a number of identities relating the sofic entropy of a certain class of non expansive algebraic dynamical systems the sofic entropy of the wired spanning forest and the tree entropy of cayley graphs of residually finite groups we also show that homoclinic points and periodic points in harmonic models are dense under general conditions

random attractors for the coupled suspension bridge equations with white noises the paper is devoted to the investigation of the existence of a compact random attractor for the random dynamical system generated by the coupled suspension bridge equations with white noises

random attractors for the extensible suspension bridge equation with white noise this paper is devoted to the dynamical behavior of a stochastic extensible suspension bridge equation with white noise for a deterministic suspension bridge equation there are many classical results such as existence and uniqueness of a solution and long term behavior of solutions to the best of our knowledge the existence of attractors for the suspension bridge equation with white noise was not yet considered we intend to investigate these problems we first obtain the dissipativeness of a solution in higher energy spaces h 3 h 2 h 0 1 this implies that the random dynamical system generated by the equation has a random attractor in h 2 h 0 1 l 2 which is a tempered random set in the space in h 3 h 2 h 0 1

global attractors for a semilinear hyperbolic equation in viscoelasticity a semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered regarding the past history memory of the displacement as a new variable the equation is transformed into a dynamical system in a suitable hilbert space the dissipation is extremely weak and it is all contained in the memory term longtime behavior of solutions is analyzed in particular in the autonomous case the existence of a global attractor for solutions is achieved

existence of global attractors for the coupled system of suspension bridge equations a mathematical model of the suspension bridge describes the vibration of the road bed in the vertical plain and that of the main cable we show the existence of an absorbing set for the solution of the problem furthermore the global attractors of the coupled system of suspension bridge are studied by a new semigroup approach

existence of strong global attractors for hyperbolic equation with linear memory we prove that the existence of global attractors of strong solutions for the hyperbolic equations with linear memory using the semigroup approach