periodically forced double homoclinic loops to a dissipative saddle abstract in this paper we present a comprehensive theory on the dynamics of strange attractors in periodically perturbed second order differential equations assuming that the unperturbed equations have two homoclinic loops to a dissipative saddle fixed point we prove the existence of many complicated dynamical objects for a large class of non autonomous second order equations ranging from attractive quasi periodic torus to newhouse sinks and henon like attractors and to rank one attractors with srb measures and full stochastic behavior this theory enables us to apply rigorously many profound dynamics theories on non uniformly hyperbolic maps developed in the last forty years including the newhouse theory the theory of srb measures the theory of henon like attractors and the theory of rank one attractors to the analysis of the strange attractors in a periodically perturbed duffing equation

compaction retraction and constraint satisfaction in this paper we show a very close relationship among the compaction retraction and constraint satisfaction problems in the context of reflexive and bipartite graphs the compaction and retraction problems are special graph coloring problems and the constraint satisfaction problem is well known to have an important role in artificial intelligence the relationships we present provide evidence that similar to as for the retraction problem it is likely to be difficult to determine whether for every fixed reflexive or bipartite graph the compaction problem is polynomial time solvable or np complete in particular the relationships that we present relate to a long standing open problem concerning the equivalence of the compaction and retraction problems

kms states and conformal measures from a non constant holomorphic map on a connected riemann surface we construct an etale second countable locally compact hausdorff groupoid whose associated groupoid c algebra admits a one parameter group of automorphisms with the property that its kms states correspond to conformal measures in the sense of sullivan in this way certain quadratic polynomials give rise to quantum statistical models with a phase transition arising from spontaneous symmetry breaking

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

on the dynamics of a family of non collet eckmann rational maps abstract considering a family of rational maps t n relating to renormalization transformations we prove that each fatou component of t n is a holder domain for odd integers n 5 and some parameters 1 2 but t n is not a collet eckmann map even if there is only one critical point in its julia set furthermore we show that there exists a buried open interval in the julia set j t n but the two endpoints of this interval belong to the boundary of some fatou component of t n

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

rational misiurewicz maps are rare we show that the set of misiurewicz maps has lebesgue measure zero in the space of rational functions for any fixed degree d 2

strange attractors in saddle node cycles prevalence and globality we consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle node cycles we show that they always exhibit henon like strange attractors for a set of parameter values with positive lebesgue density at the bifurcation value in open classes of such families the strange attractors are of global type their basins contain an a priori defined neighbourhood of the cycle furthermore the bifurcation parameter may also be a point of positive density of hyperbolic dynamics

noncommutative algebras for hyperbolic diffeomorphisms the gibbs states of classical equilibrium statistical mechanics can be extended to states on non commutative algebras satisfying the kubo martin schwinger boundary condition this way of looking at gibbs states is applied here to the study of differentiable dynamical systems when some strong or weak hyperbolicity conditions are satisfied

simulating memory effects with discrete dynamical systems abstract we propose using discrete dynamical systems to model non markovian processes this implies a whole hierarchy of dynamical systems with dimensionality increasing proportional to the memory specific long range memory effects are investigated for a quadratic a cubic and a quartic map markovian processes transform the quadratic map into a normal form of the logistic equation the henon map corresponds to non markovian first generation memory effects higher memories imply new high dimensional systems non markovian processes imply new memory routes to chaos for the three polynomial maps discussed here it is possible to define critical memory lengths above which the systems essentially lose memory

groupoids associated with endomorphisms to a compact hausdorff space which covers itself we associate an r discrete locally compact hausdorff groupoid its c algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the cuntz algebra op we consider examples related to coverings of the circle and of a heisenberg 3 manifold given a compact hausdorff space x and a continuous surjective map a x x we want to associate to the pair x a a c algebra c x a which in the case a is a homeomorphism coincides with the well known crossed product c x x z in this paper we consider the case of a covering a x x and c x a will be defined via a groupoid that generalizes the construction of the cuntz groupoid op considered by renault in re in that case x is a unilateral sequence space and a is the unilateral shift we start with some definitions and notation that make sense in greater generality definition given a set x and a map a x x let i x n y e xx zx x i 3k 1 0 n i k akx aly with the set of composable pairs i 2 x n y w m z e jx j w y the multiplication and inversion are given by x n y y m z x m n z and x n y 1 y n x the range and source maps are r x n y x 0 x s x n y y 0 y with these operations if becomes a groupoid with the unit space identified with x the equivalence relation associated to if will be denoted by r where r x y e x x x i 3k 1 0 akx ly and the isotropy group bundle is i x n x exxzxx ik 1 0 n l k akx alx received by the editors march 25 1993 and in revised form january 21 1994 originally communicated to the proceedings of the ams by palle e t jorgensen 1991 mathematics subject classification primary 46l05 46l55 46l80 research supported in part by the national science foundation 1995 american mathematical society 0002 9947 95 1 00 25 per page

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

equivalence and topological invariance of conditions for non uniform hyperbolicity in the iteration of rational maps we show equivalence of several standard conditions for non uniform hyperbolicity of complex rational functions including the topological collet eckmann condition tce uniform hyperbolicity on periodic orbits exponential shrinking of components of pre images of small discs backward collet eckmann condition at one point positivity of the infimum of lyapunov exponents of finite invariant measures on the julia set the condition tce is stated in purely topological terms so we conclude that all these conditions are invariant under topological conjugacy for rational maps with one critical point in julia set all the conditions above are equivalent to the usual collet eckmann and backward collet eckmann conditions thus the latter ones are invariant by topological conjugacy in the unicritical setting we also prove that neither part of this stronger statement is valid in the multicritical case

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

fuzzy partitions and fuzzy quotient rings abstract the cosets of a fuzzy ideal m in a ring a form another ring a m called the fuzzy quotient ring of a and m we show that a m is always a fuzzy partition of a and a fuzzy partition of a will be a fuzzy quotient ring with respect to some fuzzy ideal if and only if its associated fuzzy similarity relation is compatible with the ring operations we also discuss relationships between the two operations of the fuzzy quotient ring a m and the extensions over ia of the two operations of a by zadeh s extension principle we study relationships between the level ring structures of a fuzzy quotient ring and the level relations of its associated fuzzy similarity relations we introduce and study a new notion of the factorization of a ring homomorphism through fuzzy quotients finally a relationship between fuzzy prime ideals and fuzzy similarity relation has been established

the adjoint representation of fuzzy lie algebras the commutator of a lie algebra l is extended by zadeh s extension principle to a product of fuzzy subsets a fuzzy subspace generated by the product of two fuzzy ideals is also a fuzzy ideal this notion of product of fuzzy ideals is used to define the descending central series of a fuzzy ideal nilpotency of fuzzy ideals is introduced using this descending central series fuzzification of the adjoint representation of lie algebras is introduced and some relations between this representation and nilpotent fuzzy ideals are proved finally the killing form in the fuzzy case is defined and some of its properties are studied

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

fuzzy linear and affine spaces abstract elementary properties of fuzzy linear and affine spaces are studied the concept of a basis for a fuzzy linear space is defined and an algorithm for computing a basis when the underlying space is finite dimensional is given

fuzzy ideals and fuzzy subalgebras of lie algebras abstract a fuzzy idea l of a lie algebra l factors it into a lie algebra l l called a fuzzy quotient lie algebra this fuzzy algebra lie algebra is always a fuzzy partition of l and a fuzzy partition of l will be a fuzzy quotient lie algebra if and only if its associated fuzzy similarity relation is compatible with the lie algebra operations relationship between the operations of the lie algebras l i t and zadeh s extension principle are studied we also study homomorphisms of lie algebras and the relationship between them and fuzzy ideals and fuzzy subalgebras of the domains and the co domains of these homomorphisms finally solvability of fuzzy ideals and studied in defining these solvable fuzzy ideals we use zadeh s extension principle

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

dynamics of horizontal like maps in higher dimension we study the regularity of the green currents and of the equilibrium measure associated to a horizontal like map in under a natural assumption on the dynamical degrees we estimate the speed of convergence towards the green currents the decay of correlations for the equilibrium measure and the lyapounov exponents we show in particular that the equilibrium measure is hyperbolic we also show that the green currents are the unique invariant vertical and horizontal positive closed currents the results apply in particular to henon like maps to regular polynomial automorphisms of and to their small perturbations

duality and polynomial testing of tree homomorphisms let h be a fixed digraph we consider the h colouring problem i e the problem of deciding which digraphs g admit a homomorphism to h we are interested in a characterization in terms of the absence in g of certain tree like obstructions specifically we say that h has tree duality if for all digraphs g g is not homomorphic to h if and only if there is an oriented tree which is homomorphic to g but not to h we prove that if h has tree duality then the h colouring problem is polynomial we also generalize tree duality to bounded treewidth duality and prove a similar result we relate these duality concepts to the notion of the x property studied by gutjahr welzl and woeginger we then focus on the case when h itself is an oriented tree in fact we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two such trees are called triads we have shown in a companion paper that there exist oriented triads h for which the h colouring problem is np complete we contrast these with several families of oriented triads h which have tree duality or bounded treewidth duality and hence polynomial h colouring problems if p 6 np then no oriented triad h with an np complete h colouring problem can have bounded treewidth duality however no proof of this is known for any oriented triad h we prove that none of the oriented triads h with np complete hcolouring problems given in the companion paper has tree duality

the lyapunov exponent of holomorphic maps we prove that for any polynomial map with a single critical point its lower lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle similar statement holds for the exponential maps and some other complex dynamical systems we prove further that for the unicritical polynomials with positive area julia sets almost every point of the julia set has zero lyapunov exponent part of this statement generalizes as follows every point with positive upper lyapunov exponent in the julia set of an arbitrary polynomial is not a lebegue density point

distribution of rational maps with a preperiodic critical point let f lambda be a family of rational maps of a fixed degree with a marked critical point c lambda under a natural assumption we first prove that the hypersurfaces of parameters for which c lambda is periodic converge as a sequence of positive closed 1 1 currents to the bifurcation current attached to c and defined by demarco we then turn our attention to the parameter space of polynomials of a fixed degree d by intersecting the d 1 currents attached to each critical point of a polynomial bassaneli and berteloot obtained a positive measure mu rm bif of finite mass which is supported on the connectedness locus they showed that its support is included in the closure of the set of parameters admitting d 1 neutral cycles we show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials i e of misiurewicz points

collet eckmann and the bifurcation measure the moduli space mathcal m d of degree d ge 2 rational maps can naturally be endowed with a measure mu text bif detecting maximal bifurcations called the bifurcation measure we prove that the support of the bifurcation measure mu text bif has positive lebesgue measure to do so we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of mu text bif and we exhibit a large set of collet eckmann rational maps which satisfy this condition as a consequence we get a set of collet eckmann rational maps of positive lebesgue measure which are approximated by hyperbolic rational maps

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

sna s in the quasi periodic quadratic family we rigorously show that there can exist strange nonchaotic attractors sna in the quasi periodically forced quadratic or logistic map theta x mapsto theta omega c theta x 1 x for certain choices of c mathbb t to 3 2 4 and diophantine

unfolding a chaotic bifurcation the differential equation of the sinusoidally forced pendulum is studied by digital simulation in a regime where two simple symmetrically related chaotic attractors grow and merge continuously as the forcing amplitude is increased by introducing a small constant bias in the forcing to break the symmetry two discontinuous bifurcations unfold from the single merging event considering both forcing amplitude and bias together as controls a codimension two bifurcation of chaotic attractors is defined whose geometric structure in control phase space is closely related to the elementary cusp catastrophe the chaotic bifurcations are explained in terms of homoclinic structures smale cycles in the poincare map

on the complexity of the general coloring problem a graph h is called an interpretation of a graph g if a morphic image of h is isomorphic to a subgraph of g a graph h can be seen to be n colorable iff h is an interpretation of k n the complete graph on n vertices hence colorability is a special case of the concept of interpretation in this paper the complexity of the general coloring problem i e of deciding for a given graph g whether some graph h is an interpretation of g is investigated it is shown that for many very simple undirected graphs g this question is np complete this was previously known for the graphs k n only in fact it seems to be np complete for all but trivial exceptions in the directed case non trivial graphs g for which the problem is in p and rather simple graphs g for which the problem is np complete are presented a characterization of all graphs g for which the problem at issue is in p remains an evasive open problem

fine structure of connectedness loci we study distributions of the harmonic and lebesgue measures on the boundary of the connectedness locus mathcal m d for uni critical polynomials z d c and dynamical properties which are typical with respect to the harmonic measure one result is a conformal similarity between mathcal m d and the corresponding julia set for almost every point of mathcal m d with respect to the harmonic measure it is also shown that the harmonic measure is supported on lebesgue density points of the complement of mathcal m d which are not accessible from outside within john angles and at which the boundary of mathcal m d spirals infinitely often in both directions in the case of quadratic polynomials we prove that every parameter c from the boundary of the mandelbrot set mathcal m that is recurrent under iterates of z 2 c but not infinitely tunable is a lebesgue weak density point of the complement of mathcal m a direct consequence of our theory is the yoccoz local connectivity theorem a new technical ingredient in the paper is the method of amplification which is based on promoting the small scale geometry of the parameter space at a typical point in the sense of harmonic measure to uniform geometry of the corresponding julia set the pointwise conformality of the similarity map is derived through inducing techniques and a classical twb theory

statistical properties of the universal limit map of grazing bifurcations in this paper the statistical properties of an interval map having a square root singular point which characterizes grazing bifurcations of impact oscillators are studied firstly we show that in some parameter regions the map admits an induced markov structure with an exponential decay tail of the return times then we prove that the map has a unique mixing absolutely continuous invariant probability measure finally by applying the markov tower method we prove that exponential decay of correlations and the central limit theorem hold for holder continuous observations

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

heteroclinic tangles in time periodic equations abstract in this paper we prove that when a heteroclinic loop is periodically perturbed three types of heteroclinic tangles are created and they compete in the space of where is a parameter representing the magnitude of the perturbations the three types are a transient heteroclinic tangles containing no gibbs measures b heteroclinic tangles dominated by sinks representing stable dynamical behavior and c heteroclinic tangles with strange attractors admitting srb measures representing chaos we also prove that as 0 the organization of the three types of heteroclinic tangles depends sensitively on the ratio of the unstable eigenvalues of the saddle fixed points of the heteroclinic connections the theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are well beyond the capacity of the classical birkhoff melnikov smale method

intuitionistic fuzzy lie sub superalgebras and intuitionistic fuzzy ideals in this paper the concepts of intuitionistic fuzzy sets are introduced to lie superalgebras intuitionistic fuzzy lie sub superalgebras and intuitionistic fuzzy ideals of lie superalgebras are defined and related properties are investigated as the applications of intuitionistic fuzzy lie sub superalgebras and intuitionistic fuzzy ideals the properties of intuitionistic fuzzy lie sub superalgebras and intuitionistic fuzzy ideals under homomorphisms of lie superalgebras are studied the intuitionistic fuzzy bracket product is also introduced and its characterization is established

post critical set and non existence of preserved meromorphic 2 forms we present a family of birational transformations in cp2 depending on two or three parameters which do not generically preserve meromorphic 2 forms with the introduction of the orbit of the critical set vanishing condition of the jacobian also called the post critical set we get some new structures some non analytic 2 form which reduce to meromorphic 2 forms for particular subvarieties in the parameter space on these subvarieties the iterates of the critical set have a polynomial growth in the degrees of the parameters while one has an exponential growth out of these subspaces the analysis of our birational transformation in cp2 is first carried out using the diller favre criterion in order to find the complexity reduction of the mapping the integrable cases are found the identification between the complexity growth and the topological entropy is once again verified we perform plots of the post critical set as well as calculations of lyapunov exponents for many orbits confirming that generically no meromorphic 2 form can be preserved for this mapping these birational transformations in cp2 which generically do not preserve any meromorphic 2 form are extremely similar to other birational transformations we previously studied which do preserve meromorphic 2 forms we note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned but drastically different results as far as a more probabilistic approach of dynamical systems is concerned lyapunov exponents with these examples we see that the existence of a preserved meromorphic 2 form explains most of the numerical discrepancies between the topological and probabilistic approaches of dynamical systems

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

fuzzy quotient groups abstract the operation of a group g induces through zadeh s extension principle a semigroup operation on ig we define a fuzzy quotient group of g to be a fuzzy partition of g that is a subsemigroup of ig having a group structure we show that fuzzy quotient groups of g are in one to one correspondence with normal fuzzy subgroups of g and that their associated fuzzy similarity relations are all two sided invariant we study relationships between the level structures of these structures we also give a necessary and sufficient condition for a group homomorphism to factor through fuzzy quotients

maximal abelian subalgebras of c algebras associated with complex dynamical systems and self similar maps abstract we consider an analogy among markov shifts complex dynamical systems and self similar maps their dynamics are given by 0 1 matrices a rational functions r and self similar maps on a compact metric space k respectively if the 0 1 matrix a is irreducible and not a permutation then the cuntz krieger algebra o a is simple and purely infinite similarly if the rational function r is restricted to the julia set j r and the self similar map satisfies the open set condition respectively then the associated c algebras o r j r and o k are simple and purely infinite let a be the associated infinite path space for the 0 1 matrix a then c a is known to be a maximal abelian subalgebra of o a in this paper we shall show that c j r is a maximal abelian subalgebra of o r j r and c k is a maximal abelian subalgebra of o k

from local to global consistency abstract in reasoning tasks involving the maintenance of consistent databases so called qqconstraint networks q q it is customary to enforce local consistency conditions in order to simplify the subsequent construction of a globally coherent model of the data in this paper we present a relationship between the sizes of the variables domains the constraints arity and the level of local consistency sufficient to ensure global consistency based on these parameters a new tractability classification of constraint networks is presented we also show based on this relationship that any relation on bi valued variables which is not representable by a network of binary constraints cannot be represented by networks with any number of hidden variables

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

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

the csp dichotomy holds for digraphs with no sources and no sinks a positive answer to a conjecture of bang jensen and hell bang jensen and hell conjectured in 1990 using the language of graph homomorphisms a constraint satisfaction problem csp dichotomy for digraphs with no sources or sinks the conjecture states that the csp for such a digraph is tractable if each component of its core is a cycle and is np complete otherwise in this paper we prove this conjecture and as a consequence a conjecture of bang jensen hell and macgillivray from 1995 classifying hereditarily hard digraphs further we show that the csp dichotomy for digraphs with no sources or sinks agrees with the algebraic characterization conjectured by bulatov jeavons and krokhin in 2005

finite digraphs and kms states abstract the paper contains a description of the kms states of a generalized gauge action on the c algebra of a finite graph

potentials on the two torus for which the hamiltonian flow is ergodic we consider the motion of a test particle in a compound central potential field on a two dimensional torus we discuss three different classes of potentials attracting repelling and mixed that lead to hamiltonian systems which have positive lyapunov exponent almost everywhere and are ergodic included among the mixed potentials are smooth potentials without singularities

kms states on pimsner algebras associated with c dynamical systems abstract we examine the theory of the kms states on pimsner algebras arising from multivariable unital c dynamical systems as an application we show that pimsner algebras of piecewise conjugate classical systems attain the same kms states even though it is still an open problem whether or not they are isomorphic

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

h coloring degree bounded acyclic digraphs an np complete coloring or homomorphism problem may become polynomial time solvable when restricted to graphs with degrees bounded by a small number but remain np complete if the bound is higher we investigate an analogous phenomenon for digraphs focusing on the three smallest digraphs h with np complete h colorability problems it turns out that in all three cases the h coloring problem is polynomial time solvable for digraphs with in degrees at most 1 regardless of the out degree bound respectively with out degrees at most 1 regardless of the in degree bound on the other hand as soon as both in and out degrees are bounded by constants greater than or equal to 2 all three problems are again np complete interestingly the situation changes when we further restrict the inputs to be acyclic digraphs in two of the three cases the np complete problems remain np complete in the third case the h coloring problem turns out to be polynomial time solvable for acyclic digraphs with in degrees at most 2 regardless of the out degree bound respectively with out degrees at most 2 regardless of the in degree bound we also show that in this case as soon as both in and out degrees are bounded by constants greater than or equal to 3 the h coloring problem is once again np complete even for acyclic digraphs a conjecture proposed for graphs h by feder hell and huang states that any variant of the h coloring problem which is np complete without degree constraints is also np complete with degree constraints provided the degree bounds are high enough it has been verified for h coloring of digraphs but the degree bounds are quite high and are stated in terms of the sum of in and out degrees our results confirm the conjecture with the best possible bounds that are needed for np completeness moreover they underscore the fact that the bound must apply separately to both the in degrees and the out degrees

conjunctive queries over trees we study the complexity and expressive power of conjunctive queries over unranked labeled trees represented using a variety of structure relations such as child descendant and following as well as unary relations for node labels we establish a framework for characterizing structures representing trees for which conjunctive queries can be evaluated efficiently then we completely chart the tractability frontier of the problem and establish a dichotomy theorem for our axis relations that is we find all subset maximal sets of axes for which query evaluation is in polynomial time and show that for all other cases query evaluation is np complete all polynomial time results are obtained immediately using the proof techniques from our framework finally we study the expressiveness of conjunctive queries over trees and show that for each conjunctive query there is an equivalent acyclic positive query i e a set of acyclic conjunctive queries but that in general this query is not of polynomial size

periodicity versus chaos in one dimensional dynamics we survey recent results in one dimensional dynamics and as an application we derive rigorous basic dynamical facts for two standard models in population dynamics the ricker and the hassell families we also informally discuss the concept of chaos in the context of one dimensional discrete time models first we use the model case of the quadratic family for an informal exposition we then review precise generic results before turning to the population models our focus is on typical asymptotic behavior seen for most initial conditions and for large sets of maps parameter sets corresponding to different types of attractors are described in particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models natural measures asymptotic distributions and their stability properties are considered

hausdorff dimension of julia sets of feigenbaum polynomials with high criticality we consider unimodal polynomials with feigenbaum topological type and critical points whose orders tend to infinity it is shown that the hyperbolic dimensions of their julia set go to 2 furthermore that the hausdorff dimensions of the basins of attraction of their feigenbaum attractors also tend to 2 the proof is based on constructing a limiting dynamics with a flat critical point

abundance of mode locking for quasiperiodically forced circle maps we study the phenomenon of mode locking in the context of quasiperiodically forced non linear circle maps as a main result we show that under certain mathcal c 1 open condition on the geometry of twist parameter families of such systems the closure of the union of mode locking plateaus has positive measure in particular this implies the existence of infinitely many mode locking plateaus open arnold tongues the proof builds on multiscale analysis and parameter exclusion methods in the spirit of benedicks and carleson which were previously developed for quasiperiodic rm sl 2 mathbb r cocycles by young and bjerklov the methods apply to a variety of examples including a forced version of the classical arnold circle map