on the directional dynamics of additive cellular automata we continue the study of cellular automata ca directional dynamics i e the behavior of the joint action of ca and shift maps this notion has been investigated for general ca in the case of expansive dynamics by boyle and lind and by sablik for sensitivity and equicontinuity in this paper we give a detailed classification for the class of additive ca providing non trivial examples for some classes of sablik s classification moreover we extend the directional dynamics studies by considering also factor languages and attractors

kolmogorov complexity and cellular automata classification we present a new approach to cellular automata ca classification based on algorithmic complexity we construct a parameter which is based only on the transition table of ca and measures the randomness of evolutions is better in a certain sense than any other parameter recursively definable on ca tables we investigate the relations between the classical topological approach and ours our parameter is compared with langton s parameter turns out to be theoretically better and also agrees with some practical evidences reported in literature finally we propose a protocol to approximate and make experiments on ca dynamical behavior

computation theory of cellular automata self organizing behaviour in cellular automata is discussed as a computational process formal language theory is used to extend dynamical systems theory descriptions of cellular automata the sets of configurations generated after a finite number of time steps of cellular automaton evolution are shown to form regular languages many examples are given the sizes of the minimal grammars for these languages provide measures of the complexities of the sets this complexity is usually found to be non decreasing with time the limit sets generated by some classes of cellular automata correspond to regular languages for other classes of cellular automata they appear to correspond to more complicated languages many properties of these sets are then formally non computable it is suggested that such undecidability is common in these and other dynamical systems

tightness and distinguished frechet spaces in this chapter we apply the concept of tightness to study distinguished frechet spaces we show that a frechet space is distinguished if and only if its strong dual has countable tightness this approach to studying distinguished frechet spaces leads to a rich supply of df spaces whose weak duals are quasi suslin but not k analytic the small cardinals mathfrak b and mathfrak d will be used to improve the analysis of kothe s echelon nondistinguished frechet space 1 a

some results about the chaotic behavior of cellular automata we study the behavior of cellular automata ca for short in the cantor besicovitch and weyl topologies we solve an open problem about the existence of transitive ca in the besicovitch topology the proof of this result has some interest of its own since it is obtained by using kolmogorov complexity to our knowledge it is the first result about discrete dynamical systems obtained using kolmogorov complexity we also prove that in the besicovitch topology every ca has either a unique periodic point thus a fixed point or an uncountable set of periodic points this result underlines the fact that ca have a great degree of stability it may be considered a further step towards the understanding of ca periodic behavior moreover we prove that in the besicovitch topology there is a special set of configurations the set of toeplitz configurations that plays a role similar to that of spatially periodic configurations in the cantor topology that is it is dense and has a central role in the study of surjectivity and injectivity finally it is shown that the set of spatially quasi periodic configurations is not dense in the weyl topology

number conserving cellular automata i decidability we prove that definitions of number conserving cellular automata found in literature are equivalent a necessary and sufficient condition for cellular automata to be number conserving is proved using this condition we give a quasi linear time algorithm to decide number conservation

a quantitative approach to weak compactness in frechet spaces and spaces c x let ee be a frechet space i e a metrizable and complete locally convex space lcs e e its strong second dual with a defining sequence of seminorms n n induced by a decreasing basis of absolutely convex neighbourhoods of zero unun and let h eh e be a bounded set let ck h sup d cluste e hn ck h sup d cluste e hn be the worst distance of the set of weak cluster points in e e of sequences in hh to ee and k h sup d h e h h the worst distance of h the weak closure in the bidual of hh to ee where dd means the natural metric of e e let n h sup limplimmup hm limmlimpup hm up un0 hm h provided the involved limits exist we extend a recent result of angosto cascales to frechet spaces by showing that if x h there is a sequence xp p xp p in hh such that dn x y n h dn x y n h for each e e e e cluster point y y of xp p xp p and n nn n moreover k h 0k h 0 iff ck h 0ck h 0 this provides a quantitative version of the weak angelicity in a frechet space also we show that ck h d h c x z 17ck h where h zxh zx is relatively compact and c x z c x z is the space of zz valued continuous functions for a web compact space xx and a separable metric space zz being now ck h ck h the worst distance of the set of cluster points in zxzx of sequences in hh to c x z c x z respect to the standard supremum metric dd and d h c x z sup f c x z f h this yields a quantitative version of orihuela s angelic theorem if xx is strongly web compact then ck h d h c x z 5ck h this happens if x e e e x e e e for e ge g for instance if ee is a df space or an lf space in the particular case that ee is a separable metrizable locally convex space then d h c x z ck h for each bounded h rxh rx

the intersection of opposed big cells in real flag varieties for any reductive algebraic group over c split over r a graph is defined in type a and d it is shown that the number of connected components of the graph coincides with the number of connected components of the intersection of two opposed big cells in the corresponding real flag variety this gives a combinatorial description for the number of connected components of such a variety

the intersection of opposed big cells in the real flag variety of type g2 we compute the euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type g2 verifying a conjecture from 6

universality and complexity in cellular automata abstract cellular automata are discrete dynamical systems with simple construction but complex self organizing behaviour evidence is presented that all one dimensional cellular automata fall into four distinct universality classes characterizations of the structures generated in these classes are discussed three classes exhibit behaviour analogous to limit points limit cycles and chaotic attractors the fourth class is probably capable of universal computation so that properties of its infinite time behaviour are undecidable

weight of precompact subsets and tightness pfister 1976 and cascales and orihuela 1986 proved that precompact sets in df and lm spaces have countable weight i e are metrizable improvements by valdivia 1982 cascales and orihuela 1987 and kakol and saxon preprint have varying methods of proof for these and other improvements a refined method of upper semi continuous compact valued maps applied to uniform spaces will suffice at the same time this method allows us to dramatically improve kaplansky s theorem that the weak topology of metrizable spaces has countable tightness extending it to include all lm spaces and all quasi barrelled df spaces both in the weak and original topologies one key is showing that for a large class g including all df and lm spaces countable tightness of the weak topology of e in g is equivalent to realcompactness of the weak topology of the dual of e

chaos and diffusion in deterministic cellular automata abstract it is shown that some of the deterministic one dimensional cellular automata studied recently by wolfram exhibit a kind of spontaneous symmetry breaking the associated kinks bloch walls perform annihilating diffusive walks for infinite lattices and large times in addition close analogies between these automata and strange attractors are exhibited by mapping the automata onto dynamical systems defined by iterations of the unit square

thermodynamics of systems of finite sequences it is proposed that the hamming distance between bit strings plays the role of interaction energy the statistical sum and thermodynamic functions are calculated for the canonical ensemble of bit sequences the method is illustrated with an ensemble of two sequence systems which evolve according to cellular automata rules

universality and decidability of number conserving cellular automata number conserving cellular automata ncca are particularly interesting both because of their natural appearance as models of real systems and because of the strong restrictions that number conservation implies here we extend the definition of the property to include cellular automata with any set of states in z and show that they can be always extended to usual ncca with contiguous states we show a way to simulate any one dimensional ca through a one dimensional ncca proving the existence of intrinsically universal ncca finally we give an algorithm to decide given a ca if its states can be labeled with integers to produce a ncca and to find this relabeling if the answer is positive

a hierarchical classification of cellular automata abstract a central issue in the theory of cellular automata is classification a classification is a structure imposed on the space of automata which groups together cellular automata with related properties in this paper a classification based on the action of cellular automata on n step markov measures is presented this classification is hierarchical and parametric an algorithm is described which efficiently constructs all rules in a given class at a given level of the hierarchy the utility of the classification is explored in the study of the invariant statistical properties of next nearest neighbor cellular automata on two states per cell

on conservative and monotone one dimensional cellular automata and their particle representation number conserving or conservative cellular automata ca have been used in several contexts in particular traffic models where it is natural to think about them as systems of interacting particles in this article we consider several issues concerning one dimensional cellular automata which are conservative monotone specially non increasing or that allow a weaker kind of conservative dynamics we introduce a formalism of particle automata and discuss several properties that they may exhibit some of which like anticipation and momentum preservation happen to be intrinsic to the conservative ca they represent for monotone ca we give a characterization and then show that they too are equivalent to the corresponding class of particle automata finally we show how to determine for a given ca and a given integer b whether its states admit a b neighborhood dependent relabeling whose sum is conserved by the ca iteration this can be used to uncover conservative principles and particle like behavior underlying the dynamics of some ca compliments at http www dim uchile cl anmoreir ncca

classifying circular cellular automata we introduce a hierarchy of linear cellular automata based on their limiting behavior on spatially periodic configurations we show that it is undecidable to which class in the hierarchy a cellular automaton belongs in particular it is undecidable whether all spatially periodic configurations evolve to a fixed point furthermore there is no computable bound on the period lengths of these configurations our arguments are based on a non standard simulation of turing machines on circular cellular automata

generalized sub shifts in elementary cellular automata the strange case of chaotic rule 180 abstract we study the dynamical behavior of elementary cellular automaton 180 this rule gives rise to a global dynamics on the phase space of all one dimensional bi infinite configurations which is devaney topologically chaotic the dense sub dynamical system of configurations in background of 0s is a generalized sub shift i e multiple sub shift whose multiplicity constant depends on the initial configuration this sub dynamical system is deeply stable in the sense that the null configuration is a global attractor but with some components of the chaotic behaviour transitivity and unpredictability the dense sub dynamical system of configurations in background of 1s is a fractal like system with strongly chaotic components expansivity

weakly k analytic spaces and the three space property for analyticity abstract let e e be a dual pair of vector spaces the paper studies general conditions which allow to lift analyticity or k analyticity from the weak topology e e to stronger ones in the frame of e e first we show that the mackey dual of a space c p x is analytic iff the space x is countable this yields that for uncountable analytic spaces x the mackey dual of c p x is weakly analytic but not analytic we show that the mackey dual e of an lf space of a sequence of reflexive separable frechet spaces with the heinrich density condition is analytic i e e is a continuous image of the polish space n n this extends a result of valdivia we show also that weakly quasi suslin locally convex baire spaces are metrizable and complete this extends a result of de wilde and sunyach we provide however a large class of weakly analytic but not analytic metrizable separable baire topological vector spaces not locally convex this will be used to prove that analyticity is not a three space property but we show that a metrizable topological vector space e is analytic if e contains a complete locally convex analytic subspace f such that the quotient e f is analytic several questions remarks and examples are included

two dimensional cellular automata a largely phenomenological study of two dimensional cellular automata is reported qualitative classes of behavior similar to those in one dimensional cellular automata are found growth from simple seeds in two dimensiona cellular automata can produce patterns with complicated boundaries characterized by a variety of growth dimensions evolution from disordered states can give domains with boundaries that execute effectively continuous motions some global properties of cellular automata can be described by entropies and lyapunov exponents others are undecidable

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

number conserving cellular automata ii dynamics in this second part we study the dynamics of the number conserving cellular automata we give a classification which focuses on pattern divergence and chaoticity moreover we prove that in the case of number conserving cellular automata surjectivity is equivalent to regularity as a byproduct we obtain a strong characterization of the class of cellular automata with bounded evolutions on finite configurations

quasi suslin weak duals abstract cascales kakol and saxon cks ushered kaplansky and valdivia into the grand setting of cascales orihuela spaces e by proving k if e is countably tight then so is the weak space e e e and v e e e is countably tight iff weak dual e e e is k analytic the ensuing flow of quasi suslin weak duals that are not k analytic a la valdivia s example continues here where we argue that locally convex spaces e with quasi suslin weak duals are k v s best setting largest by far optimal vis a vis valdivia the vaunted cks setting proves not larger in fact than kaplansky s we refine and exploit the quasi lb strong dual interplay

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

computational mechanics of cellular automata an example abstract we illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the space time behavior of elementary one dimensional cellular automaton rule 54 the dominant regular domain of the cellular automation is identified and a domain filter is constructed to locate and classify defects in the domain the primary particles are identified and a range of interparticle interactions is studied the deterministic equation of motion of the filtered space time behavior is derived filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higher level defects particle interactions and secondary domains we define the emergence time at which the space time behavior condenses into configurations consisting only of domains particles and particle interactions taken together these techniques serve as the basis for the investigation of pattern evolution and self organization in this representative system

pattern growth in elementary cellular automata a classification of elementary cellular automata ca based on their pattern growth is introduced it is shown that this classification is effective that is there exists an algorithm to determine to which class a given ca belongs this algorithm is based on the properties of the local rule of cas not requiring the observation of their evolution furthermore necessary and sufficient conditions to detect all the elementary cas exhibiting a shift like behavior are given these cas have interesting dynamical properties and chaotic characteristics

cellular automata in fuzzy backgrounds abstract the main purpose of this work is to understand some limitations introduced by the classical definitions of cellular automata ca to this end we have defined a new model of cas fuzzy ca s which allows the observation of interesting chaotic properties of elementary cas to date neither a formal nor a precise definition of chaos in cas exists we believe that the proposed model provides a sharper tool to detect which properties can be associated to a chaotic behavior we also define a measure rule entropy which gives information about the ca s dynamics solely on the basis of the rule table and provides theoretical explanations to some of the empirical observations

the magic rule spaces of neural like elementary cellular automata abstract a subclass of elementary cellular automata ca equivalent to a particular class of neural networks nn is studied the nn are dependent on four real parameters a b c and the subspace of the rule space which is obtained by varying the parameters and the dynamical characteristics of the rules belonging to this subset are determined finally a few nice and magic regularities in this subset are noted and explained

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

invariant strings and pattern recognizing properties of one dimensional cellular automata a cellular automaton is a discrete dynamical system whose evolution is governed by a deterministic rule involving local interactions it is shown that given an arbitrary string of values and an arbitrary neighborhood size representing the range of interaction a simple procedure can be used to find the rules of that neighborhood size under which the string is invariant the set of nearestneighbor rules for which invariant strings exist is completely specified as is the set of strings invariant under each such rule for any automaton rule an associated filtering rule is defined for which the only attractors are spatial sequences consisting of concatenations of invariant strings a result is provided defining the rule of minimum neighborhood size for which an arbitrarily chosen string is the unique invariant string the applications of filtering rules to pattern recognition problems are discussed