homoclinic bifurcations in n dimensional endomorphisms due to expanding periodic points the study of bifurcation mechanisms which determine the transition from regular to chaotic behaviour or route to chaos in endomorphisms maps with a nonunique inverse has reached a good level of theoretical knowledge for one dimensional maps we refer to the book by mira l as a review of the main results in this research area the situation is quite different as regards endomorphisms of dimensions higher than one a few results have been worked out until now 2 81 mainly in the simplest cases of two dimensional endomorphisms however many mathematical models of interest in the applications are ultimately described by maps with a nonunique inverse clearly in the plane ir2 or in higher dimensional spaces we can no longer take advantage of the koenigs lemeray graphical construction and of the qualitative behaviour of the graphs of powers of t x t x t3 x which are powerful tools to identify the local and global bifurcations of one dimensional endomorphisms x t x x e ir however to study bifurcation mechanisms in ir n 2 2 we make use of properties which are the natural generalizations of those used in the one dimensional case indeed a basic tool for studying the dynamics and bifurcation mechanisms of endomorphisms is provided by the analysis of the critical manifolds first introduced by gumowski and mira in 1965 see 2 and references therein the critical manifolds are the natural generalization to ir n 2 2 of the local extrema also called critical points of rank l of one dimensional endomorphisms in this work we deal with homoclinic bifurcations in endomorphisms of ir n 1 l due to expanding periodic points as in one dimensional endomorphisms we will see that critical manifolds play a pre eminent role our purpose is to contribute to the understanding of the bifurcation mechanisms related to homoclinic orbits occurring in n dimensional endomorphisms mechanisms strictly related to the nonuniqueness of the inverse function that is typical of endomorphisms not present in invertible maps although similar to those occurring in these maps in particular this work concerns the global bifurcation homoclinic bifurcations occurring when a repelling fixed point becomes a snap back repeller t sbr henceforth and in general when there exists a critical orbit homoclinic to an expanding

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

flat g bundles with canonical metrics on considere la relation entre la theorie invariante des actions de groupes algebriques semi simples sur des varietes algebriques complexes et le comportement de l application de moment et des donnees quantiques fournies par une structure symplectique associee on etudie un exemple de dimension infinie comprenant des fibres plats sur des varietes de riemann compactes

sur la convexite holomorphe des revetements lineaires reductifs d une variete projective algebrique complexe let x be a smooth connected compact projective variety over we study the shafarevich conjecture concerning holomorphic convexity along the lines of kollar s approach when the fundamental group of x admits large finite dimensionnal representations we prove that given n the topological covering space widetilde x n of a projective algebraic compact complex manifold x corresponding to the intersection of the kernels of all linear reductive representations of 1 x to gln is holomorphically convex in the surface case this is a corollary of a theorem due to katzarkov and ramachandran

long tailed maps as a representation of mixed mode oscillatory systems mixed mode oscillatory mmo systems are known to exhibit generic features such as the reversal of period doubling sequences and crossover to period adding sequences as bifurcation parameters are varied in addition they exhibit a nearly one dimensional unimodal poincare map with a long tail the numerical results of a map with a unique critical point map l show that these dynamical features are reproduced we show that a few generic conditions extracted from the map l are adequate to explain the reversal of period doubling sequences and crossover to period adding sequences we derive scaling relationsthat determine the parameter widths of the dominant windows of periodic orbits sandwiched between two successive states of rlk sequence and verify the same with the map l as the conditions used to derive the scaling relations do not depend on the form of map we suggest that the analysis is applicable to a family of two parameter one dimensional maps that satisfy these conditons

annular chaotic areas on etudie l existence et les proprietes d acier chaotiques annulaires pour l application t de r 2 sur lui meme t x y f x y g x y

fractional iterates for n dimensional maps this paper constitutes an extension of results of the paper by narayaninsamy int j bifurcations chaos 6 1 1996 55 67 we study here the solutions of the problem of the fractional iteration for n dimensional maps

bifurcation phenomena near homoclinic systems a two parameter analysis the bifurcations of periodic orbits in a class of autonomous three variable nonlinear differential equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues i where 1 sil nikov s condition are studied in a two parameter space the perturbed homoclinic systems undergo a countable set of tangent bifurcations followed by period doubling bifurcations leading to periodic orbits which may be attractors if 1 2 the accumulation rate of the critical parameter values at the homoclinic system is exp 2 a global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity a countable set of these cusp bifurcations with scaling properties related to the eigenvalues i of the stationary state are shown to occur in infinitely contractive flows in the two parameter space the periodic orbit attractor domain exhibits a spiral structure globally around the set of homoclinic systems in which all the different periodic orbits are continuously connected

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

linear shafarevich conjecture in this paper we settle armatively shafarevich s uniformization conjecture for varieties with linear fundamental groups we prove the strongest to date uniformization result the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex the proof is based on both known and newly developed techniques in non abelian hodge theory

synthesis of multivalued multithreshold functions for ccd implementation basic multivalued building blocks constructed using ccd charge coupled devices technology are presented they are used to realize simple threshold functions existing techniques for decomposition of multivalued multithreshold functions into simpler subfunctions are reviewed usage of the ccd technology in implementation of these subfunctions is discussed two new decomposition techniques are proposed their aim is to obtain realizations that are better suited to ccd technology in terms of the maximum number of thresholds per element and the generation of negative weights synthesis techniques based on the proposed decompositions are presented and compared although 4 valued logic functions are stressed the techniques are in principle applicable to functions of any radix

a method to construct irregular fractal curves abstract in this paper we present a method of constructing irregular fractal curves this method is based on a sequence n n n of piecewise differentiable maps defined succinctly in narayaninsamy t narayaninsamy on basin boundaries appl math comput 99 1999 261 274 the experimental results concerning the basins of attraction of attractors which are given in gardini mira et al and narayaninsamy l gardini some global bifurcations of two dimensional endomorphisms by use of critical lines nonlinear anal theory meth appl 18 4 1991 361 399 c mira d fournier prunaret l gardini h kawakami j c cathala basin bifurcations of two dimensional noninvertible maps fractalization of basins j bifurcat chaos 2 1994 343 381 t narayaninsamy contribution a l etude de l iteration fractionaire et a celle des endomorphismes bi dimensionnels doctorat mathematiques appliquees de l universite paul sabatier toulouse iii constitute the basic tool of this sequence

automated synthesis of combinational logic using theorem proving techniques recent findings indicate that logic design still constitutes 50 percent of the total design effort and yet few automated tools are being used in industry the use of an automated theorem proving system in the logic design process presents an intriguing addition to traditional design automation tools this paper deals with the automated synthesis of combinational logic the method consists of a theorem proving implementation of a systematic uniform procedure for the synthesis of an arbitrary switching function any multiple valued including binary function can be synthesized in a top down fashion using the functional blocks of the designer s choice the method is general enough to allow for the choice of an arbitrary logic system and radix additional constraints of modularity technology dependence fault tolerance and others may be imposed upon the design it may also be possible to accommodate into this approach formal design verification design for testability functional level modeling and formal analysis of race and hazard conditions

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

a method for the generation and processing of dyadic indexed data the elements of an array of 2n data are indexed so that adjacent elements have hamming distance 1 based on this indexing a class of invertible fast in place transformations is developed the transform coefficients depend on the hamming distance between the indexes of input data a class of similar transformations exists in the walsh hadamard domain three transform operations are discussed in detail dyadic shift transform a subsuming and averaging transformation and a weighting operation applications of the method are in the processing of boolean and fuzzy switching functions image analysis etc

cost reduction in the ccd realization of mvmt functions the cost effectiveness of using an overflow circuit for truncated difference operation and a metal line for charge replication in the charge coupled device ccd realization of multivalue multithreshold mvmt functions is considered it is shown that the use of these two circuits leads to improved realizations compared to those achieved by previous approaches

snap back repellers and chaotic attractors when homoclinic orbits to an expanding periodic point exist the point is called a snap back repeller here we consider the two dimensional piecewise linear map in canonical form continuous and discontinuous showing how snap back repellers may be associated with robust chaotic attracting sets not only with chaotic repellers examples are given both for the continuous and discontinuous maps

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

discrete dichotomies and bifurcations from critical homoclinic orbits abstract perturbed discrete systems likexn 1 f xn g xn xn r n n z when the associated unperturbed map 0 is not invertible and has acriticalorbit n homoclinic to a hyperbolic fixed pointpare studied bycriticalwe mean that thef n are invertible for any integern 0 butf 0 is not invertible the main goal is to give sufficient conditions for a bifurcationfrom zero to many homoclinicswhen the parameter crosses zero we also give a melnikov like result assuring thepersistence of homoclinicsin a complete neighborhood of 0 this result is similar to the ones obtained for diffeomorphisms and flows

the prospects for multivalued logic a technology and applications view advances in multiple valued logic mvl have been inspired in large part by advances in integrated circuit technology multiple valued logic has matured to the point where four valued logic is now part of commercially available vlsi ic s besides reduction in chip area mvl offers other benefits such as the potential for circuit test this paper describes the historical and technical background of mvl and areas of present and future application it is intended as well to serve as a tutorial for the nonspecialist

design of a multiple valued systolic system for the computation of the chrestenson spectrum this correspondence deals with the computation of the chrestenson spectrum of an n ary n place p valued function by means of a systolic system the design of a systolic system in a multiple valued environment is discussed in details and some aspects are compared to binary realizations

synthesis of discrete functions using i 2 l technology algebraic methods for designing multiple valued i2l circuits are presented a new operation namely the truncated difference is introduced and proved to be an efficient tool for both analyzing and synthesizing circuits

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

periodic solutions and chaos in a delay differential equation modelling haematopoesis simple mathematical models have been devised to describe the process of haematopoesis the formation and differentiation of blood cells within the human body l 21 haematopoesis is an example of a biological control system with an inherent delay in it this is due to the time required for primitive blood cells to mature such a delayed response is known to cause oscillatory behaviour in control systems and it has been found that similar oscillatory instabilities in blood cell counts are associated with diseases such as aplastic anaemia 2 this suggest that the study of such equations may be of biological importance mathematically they are of interest because they can exhibit complex aperiodic behaviour a model put forward for consideration by mackey glass 3 is

on euler characteristic and fundamental groups of compact manifolds let m be a compact riemannian manifold pi widetilde m rightarrow m be the universal covering and omega be a smooth 2 form on m with pi omega cohomologous to zero suppose the fundamental group pi 1 m satisfies certain radial quadratic resp linear isoperimetric inequality we show that there exists a smooth 1 form eta on widetilde m of linear resp bounded growth such that pi omega d eta as applications we prove that on a compact kahler manifold m omega with pi omega cohomologous to zero if pi 1 m is mathrm cat 0 or automatic resp hyperbolic then m is kahler non elliptic resp kahler hyperbolic and the euler characteristic 1 frac dim mathbb r m 2 chi m ge 0 resp 0

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

a connection between fractional iteration and graph theory we consider the problem of the fractional iteration for n dimensional maps we have two objectives in this paper the first objective is to establish a connection between fractional iteration and graph theory the second objective is to make a study concerning the problem of fractional iteration for n dimensional maps

sur la rigidite de certains groupes fondamentaux l arithmeticite des reseaux hyperboliques complexes et les faux plans projectifs the motivation of this work comes from the study of lattices in real simple lie groups the famous margulis s superrigidity theorem claims that finite dimensional reductive representations of any lattice of a real simple lie group of real rank 2 are superrigid as a corollary such a lattice is arithmetic these results extend to the real rank one case for lattices in sp n 1 and f4 20 by the work of corlette and gromov schoen on the other hand mostow and deligne mostow exhibited arithmetic lattices with non superrigid representations as well as non arithmetic lattices in the unitary group pu 2 1 a natural question is then to find simple sufficient conditions for superrigidity or arithmeticity of lattices in pu 2 1 rogawski conjectured the following let be a torsion free cocompact lattice in pu 2 1 such that the hyperbolic quotient m b2 verifies the cohomogical conditions b1 m 0 and h1 1 m h2 m then is arithmetic in this paper we consider a smooth complex projective surface m verifying the above cohomological assumptions and study zariski dense representations of the fundamental group 1 m in a simple k group h of k rank 2 where k denotes a local field our main result states that there are strong restrictions on such representations especially when k is non archimedean theorem 5 we then consider some cocompact lattices in pu 2 1 of special geometric interest recall that a fake p2 is a smooth complex surface distinct from p2 having the same betti numbers as p2 fake p2 exist by a result of mumford and are complex hyperbolic quotients h2 by yau s proof of the calabi conjecture they obviously verify the hypotheses of rogawski s conjecture in this case we prove that every zariski dense representation of in pgl 3 is superrigid in the sense of margulis theorem 3 as a corollary every fake p2 is an arithmetic quotient of the ball b2