critical probabilities for diversity and number of clusters in randomly occupied square lattices we report the occurrence of critical probabilities associated with the maximum of diversity and the maximum number of fragments clusters on a two dimensional square lattice some scaling relations of these two variables are observed in accordance with work on fragmentation processes

dimension of the boundary of quasiconformal siegel disks we study quasiconformal siegel disks with critical points in their boundaries the main result asserts that every subarc of the boundary of the siegel disk has the hausdorff dimension strictly larger than 1 and that the boundary does not have a tangent at any point

geometrically finite groups patterson sullivan measures and ratner s ridigity theorem the rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated ratner s theorem generalizes to these foliations with respect to the patterson sullivan measure in the spirit of mostow we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups finally we show that the frame flow on geometrically finite hyperbolic manifolds is bernoulli

effect of noise on the dynamics of a complex map at the period tripling accumulation point as shown recently o b isaeva et al phys rev e 64 055201 2001 the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems we study scaling regularities associated with the effect of additive noise upon the period tripling bifurcation cascade generalizing the renormalization group approach of crutchfield et al phys rev lett 46 933 1981 and shraiman et al phys rev lett 46 935 1981 originally developed for the period doubling transition to chaos in the presence of noise the universal constant determining the rescaling rule for the intensity of the noise in period tripling is found to be gamma 12 206 640 9 em leader numerical evidence of the expected scaling is demonstrated

universal scaling behaviour for iterated maps in the complex plane according to the theory of schroder and siegel certain complex analytic maps possess a family of closed invariant curves in the complex plane we have made a numerical study of these curves by iterating the map and have found that the largest curve is a fractal when the winding number of the map is the golden mean the fractal curve has universal scaling properties and the scaling parameter differs from those found for other types of maps also for this winding number there are universal scaling functions which describe the behaviour asn of theq n th iterates of the map whereq n is thenth fibonacci number

rigidity of time changes for horocycle flows soit t t un flot preservant la mesure sur un espace de probabilite x on etudie la rigidite des changements de t pour des lots horocycles

periodic and chaotic behaviour in a reduction of the perturbed korteweg de vries equation the dynamical behaviour of a reduction of the forced and damped korteweg de vries equation is studied numerically chaos arising from subharmonic instability and homoclinic crossings are observed both period doubling bifurcations and the melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour here characterised by a positive lyapunov exponent supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits and their behaviour under perturbation using melnikov functions

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

analytic approach to bifurcation cascades in a class of generalized henon heiles potentials we investigate the bifurcation cascades of a linear librational orbit in a generalized class of henon heiles potentials the stability traces of the new orbits created at its bifurcations are found numerically to intersect linearly at the saddle energy e 1 forming what we term the henon heiles fans in the limit close to the saddle energy e 1 where the dynamics is nearly chaotic we derive analytical asymptotic expressions for the stability traces of both types of orbits and confirm the numerically determined properties of the generalized henon heiles fans as a bonus of our results we obtain analytical approximations for the bifurcation energies e n which become asymptotically exact for e n 1

fractality of deterministic diffusion in the nonhyperbolic climbing sine map abstract the nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter we show that on a suitable coarse scale this map generates an oscillating parameter dependent diffusion coefficient similarly to hyperbolic maps whose asymptotic functional form can be understood in terms of simple random walk approximations on finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter by using a green kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized takagi functions obeying generalized de rham type functional recursion relations we finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive and that for normal diffusion it increases by increasing the parameter value

topological equivalence of flows on homogeneous spaces and divergence of one parameter subgroups of lie groups soient et des treillis et et des sous groupes a un parametre des groupes de lie connexes g et g si une des conditions a b ou c est vraie alors le theoreme a etablit que si les flots induits sur les espaces homogenes g et g sont topologiquement equivalents alors ils sont topologiquement equivalents par une application affine

scales interference and identification at the transition to chaos abstract the scaling at the transition to chaos is described by a linear autoregressive model the identification of the universal scales from experimental data generally suffers from the competition of statistical and model errors due to the presence of noise and scaling corrections respectively the dilemma is resolved through a two step least mean squares method as an application the parameter scaling data of hao and zhang j stat phys 28 1982 769 for the period doubling route to chaos of the forced brusselator are reanalysed combined with order detection methods the algorithm also provides a powerful tool to detect and analyse scales interference phenomena

transition to criticality in circle maps at the golden mean the one dimensional circle map is frequently used to model the quasiperiodic route to chaos in this paper we study the transition to criticality for circle maps with a golden mean winding number in the limit as the order of the map inflection point n 1 e approaches unity the solution to the relevant pair of functional equations consists of the subcritical solution together with a perturbative correction of order e we derive functional relations that the perturbative correction must satisfy and produce numerical evidence that these equations are indeed satisfied a similar analysis is performed for the golden mean eigenvalue equations power law conjugacy of the subcritical solution to order n 1 e enables us to obtain a closed form solution for the perturbative correction that satisfies the constraint relations exactly this is compared with the more interesting critical case associated with a circle map whose inflection point is infinitesimally close to unity

cluster size entropy in the axelrod model of social influence small world networks and mass media we study the axelrod s cultural adaptation model using the concept of cluster size entropy s c which gives information on the variability of the cultural cluster size present in the system using networks of different topologies from regular to random we find that the critical point of the well known nonequilibrium monocultural multicultural order disorder transition of the axelrod model is given by the maximum of the s c q distributions the width of the cluster entropy distributions can be used to qualitatively determine whether the transition is first or second order by scaling the cluster entropy distributions we were able to obtain a relationship between the critical cultural trait q c and the number f of cultural features in two dimensional regular networks we also analyze the effect of the mass media external field on social systems within the axelrod model in a square network we find a partially ordered phase whose largest cultural cluster is not aligned with the external field in contrast with a recent suggestion that this type of phase cannot be formed in regular networks we draw a q b phase diagram for the axelrod model in regular networks

classification of joinings for kleinian groups we classify all locally finite joinings of a horospherical subgroup action on g when g psl2 r or psl2 c and is a geometrically finite zariski dense subgroup of g this generalizes ratner s 1983 joining theorem for the case when is a lattice in g

semiclassical trace formulas for pitchfork bifurcation sequences in non integrable hamiltonian systems with mixed phase space and discrete symmetries sequences of pitchfork bifurcations of periodic orbits pave the way from integrability to chaos in extending the semiclassical trace formula for the spectral density we develop a uniform approximation for the combined contribution of pitchfork bifurcation pairs for a two dimensional double well potential and the familiar h enon heiles potential we obtain very good agreement with exact quantum mechanical calculations we also consider the integrable limit of the scenario which corresponds to the bifurcation of a torus from an isolated periodic orbit for the separable version of the h enon heiles system we give an analytical uniform trace formula which also yields the correct harmonic oscillator su 2 limit at low energies and obtain excellent agreement with the slightly coarse grained quantum mechanical density of states

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

an iterated implicit complex map abstract iterated complex maps defined by a root of a biquadratic equation g z z 0 are studied numerical studies suggest conjugacy functions with differentiable natural boundaries and maps having island chains occupying much of the riemann sphere an analogy with hamiltonian systems is discussed

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

some comments on the non self dual nahm equations some empirical results on the cubic algebra 2a i sum limits j a j a j a i are presented the algebra is satisfied at the residue of any pole in a solution to nahm s non self dual equations

processes disjoint from weak mixing we show that the family 7 of ergodic measure preserving transformations which are disjoint from every weakly mixing m p t properly contains the family of distal ergodic m p t in the topological case we show that 37j the family of proximally isometric flows is properly contained in the family f 7 of multipliers for 7

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

on the appearance of multiple attractors in discrete food chains abstract klebanoff and hastings a klebanoff a hastings chaos in three species food chains journal of mathematical biology 32 1994 427 451 detected cases of multiple attractors in continuous food chains in this paper we discuss similar phenomena in the discrete food chains introduced by lindstrom t lindstrom on the dynamics of discrete food chains low and high frequency behavior and chaos journal of mathematical biology 45 2002 396 418 the results imply that the dynamical properties including species persistence may change due to disturbances that do not involve changes in the environmental parameters thus there are possibilities that species may be eradicated or start to oscillate at different frequencies without any changes in the environment since this is now shown to hold in both seasonal and non seasonal environments we expect that this is a rather general property of ecosystems

on measures invariant under tori on quotients of semisimple groups we classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple s algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces a torus a homogeneous space on which the measure is up to nite index the haar measure a product of homogeneous spaces on each of which the action degenerates to a rank one action and a homogeneous space in which every element of the action acts with zero entropy

similarity between the mandelbrot set and julia sets the mandelbrot setm is self similar about any misiurewicz pointc in the sense that if we examine a neighborhood ofc inm with a very powerful microscope and then increase the magnification by a carefully chosen factor the picture will be unchanged except for a rotation the corresponding julia setj c is also self similar in the same sense with the same magnification factor moreover the two setsm andj c are similar in the sense that if we use a very powerful microscope to look atm andj c both focused atc the structures we see look like very much the same

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

scaling properties of ℤk 1 actions on the circle we derive universal scaling properties for k 1 actions on the circle whose generators have rotation numbers algebraic of degreek as fork 2 these properties can be explained for arbitraryk in terms of a renormalization group transformation it has at least one trivial fixed point corresponding to an action whose generators are pure rotations the spectrum of the linearized transformation in this fixed point is analyzed completely the fixed point is hyperbolic with a k 1 dimensional unstable manifold in the casek 2 the known results are therefore recovered

a path integral for turbulence in incompressible fluids in this paper a classical path integral is formulated for incompressible fluids that evolve according to the navier stokes equation the path integral propagates probability distributions deterministically on the space gvol of solenoidal velocity fields we construct a set of isp 2 charges associated with the geometry of gvol and its poisson structure and a pair of supersymmetry charges connected with the hamiltonian these charges generate exact symmetries of the classical path integral when the viscosity is set equal to zero when the effect of dissipation is included the charges associated with the poisson structure and the hamiltonian are no longer conserved charges that generate kolmogorov scaling and galilean transformations are also constructed the classical path integral is formulated in terms of vorticity as well

siegel disc singularity spectra the author numerically determines the generalized dimensions and spectrum of singularities of a critical point orbit on the boundary of a siegel disc the local scaling exponent alpha mn and spectrum dependence on the degree d of the critical point are also considered he finds that limd to infinity mod alpha mn mod d approximately 0 523 53 and hence asymptotically the lower limit of the spectrum is constant whilst the upper limit is proportional to d

cluster diversity and entropy on the percolation model the lattice animal identification algorithm we present an algorithm to identify and count different lattice animals la s in the site percolation model this algorithm allows a definition of clusters based on the distinction of cluster shapes in contrast with the well known hoshen kopelman algorithm in which the clusters are differentiated by their sizes it consists in coding each unit cell of a cluster according to the nearest neighbors nn and ordering the codes in a proper sequence in this manner a la is represented by a specific code sequence in addition with some modification the algorithm is capable of differentiating between fixed and free la s the enhanced hoshen kopelman algorithm j hoshen m w berry and k s minser phys rev e 56 1455 1997 is used to compose the set of nn code sequences of each cluster using monte carlo simulations on planar square lattices up to 2000x2000 we apply this algorithm to the percolation model we calculate the cluster diversity and cluster entropy of the system which leads to the determination of probabilities associated with the maximum of these functions we show that these critical probabilities are associated with the percolation transition and with the complexity of the system

scaling and critical probabilities for cluster size and la diversity on randomly occupied square lattices using monte carlo simulations we report the behaviour of the total number of clusters the cluster size diversity and the lattice animals la diversity on randomly occupied square lattices the critical probability associated with the maximum of these variables is determined in comparison with the percolation probability pc our simulations indicate that pc and the critical probability of the maximum cluster size diversity pc d s max occur at the same point as indicated in a previous paper tsang i r and tsang i j 1997 j phys a math gen 30 l239 the probability for the maximum number of clusters is obtained at a lower value pc n max 0 27 0 01 we describe the cluster identification algorithm used to count the different la and determined the critical probability for the maximum la diversity pc d f max 0 45 0 02 we derive the exponents characterizing the relation between d s max d f max and n max and several scaling relations between the variables measured the lattice size and the probability of occupation p in addition we show the scaling behaviour of la diversity versus cluster size diversity for each value of p

universal scalings of universal scaling exponents in the last decades renormalization group rg ideas have been applied to describe universal properties of different routes to chaos quasi periodic period doubling or tripling siegel disc boundaries etc each of the rg theories leads to universal scaling exponents which are related to the action of certain rg operators the goal of this announcement is to show that there is a principle that organizes many of these scaling exponents we give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations these relations are determined by combinatorial properties of the route and become exact in an appropriate limit

on the scaling ratios for siegel disks the boundary of the siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self similar with a certain scaling ratio the restriction of the dynamics of the quadratic polynomial to the boundary of the siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number the geometry of this self similarity is universal for a large class of holomorphic maps a renormalization explanation of this universality has been proposed in the literature

formation and maintenance of complex systems a complex system is often identified by the absence of a characteristic length e g as in a fractal a very large system subject to a fragmentation and or aggregation dynamics passes through such complex configurations we study statistically creation and maintenance of such configurations in space dimensions d 1 to 5 and find that they are easily created maintained for small large d an intermediate d such as d 3 seems to be ideal for the creation and maintenance of complex systems this has consequences in a statistical description of the universe

algebraic calculation of stroboscopic maps of ordinary nonlinear differential equations abstract the relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems maps are well adopted for numerical computation and several universal properties of them are known therefore some perturbation methods have been proposed to deduce them for physical systems which can be modeled by an ordinary differential equation ode with a small nonlinearity an iterative rigorous algebraic method for the calculation of the coefficients of a taylor expansion of a stroboscopic map from odes with not necessarily small nonlinearities is presented it is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ode is a polynomial in the state variables and if the ode has a fixed point at the origin for several nonlinear systems approximations of different orders are investigated

from mandelbrot like sets to arnold tongues a transition from mandelbrot like sets to arnold tongues is characterized via a coupling of two non identical quadratic maps proposed by us a two dimensional parameter space considering the parameters of the individual quadratic maps was used to demonstrate numerically the event the location of the parameter sets where naimark sacker bifurcations occur which is exactly the place where arnold tongues of arbitrary periods are born was computed analytically

effect of noise on the critical golden mean quasiperiodic dynamics in the circle map abstract scaling regularities are examined associated with effect of additive noise on a critical circle map at the golden mean rotation number we present an improved numerical estimate for the scaling constant of hamm and graham phys rev a46 1992 6323 responsible for the effect of noise 2 3061852653 decrease of the noise amplitude by this number ensures possibility to distinguish one more level of fractal like structure associated with increase of characteristic time scale by the golden mean factor 5 1 2 numeric results demonstrating evidence of the expected scaling are presented e g portraits of the noisy attractors devil s staircase plots lyapunov charts on the parameter plane in different scales

on the game of life population and its diversity abstract one of the most important features of biological life in all levels is its astounding diversity in this work we study the well known game life due to conway analysing the statistics of cluster population n t and cluster diversity d t we have performed simulations on life for dimensions d 1 and 2 starting with an uncorrelated distribution of live and dead sites at t 0 for d 2 we study the effect of different neighbourhood relations in identifying and counting clusters an interesting scaling relation connecting the maxima of n t and d t is found

a method for accurate stability bounds in a small denominator problem the author considers the problem of obtaining realistic lower bounds for the siegel radius recent advances of the analysis of siegel disks allows one to give a very accurate numerical algorithm based on rigorous results he finds that for non quadratic polynomial maps the maximal siegel radius might correspond to rotation numbers different from the golden mean

measurable rigidity of actions on infinite measure homogeneous spaces ii we consider the problems of measurable isomorphisms and join ings measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces our main focus is on systems of algebraic origin actions of lattices and other discrete subgroups g on homoge neous spaces g h where h is a sufficiently rich unimodular subgroup in a semi simple group g we also consider actions of discrete groups of isometries isom x of a pinched negative curvature space x acting on the space of horospheres hor x for such systems we prove that the only measurable isomorphisms joinings quotients etc are the obvious algebraic or geometric ones this work was inspired by the previous work of shalom and steger but uses completely different techniques which lead to more general results 1 introduction and statement of the main results

numerical simulations for fragmentation dynamics a geometrical model for the study of fragmentation dynamics is introduced detailed statistical analysis of the fragment distribution for percolation clusters sierpinski carpets and menger sponges on square and cubic lattices is given and static and dynamic critical exponents are calculated

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

measurable quotients of unipotent translations on homogeneous spaces let u be a nilpotent unipotent subgroup of a lie group g and let f be a closed subgroup of g marina ratner showed that every ergodic u invariant probability measure on the homogeneous space f g is of a simple algebraic form we use this fundamental new result to show that every measurable quotient of the u action on j g is of a simple algebraic form roughly speaking any quotient is a double coset space a g k

multiple mixing for a class of conservative surface flows arnold and kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established nor disproved under suitable arithmetic conditions on their unique rotation vector of full lebesgue measure in the first case and of full hausdorff dimension in the second we show that these flows are mixing of any order for this we show that they display a generalization of the so called ratner property on slow divergence of nearby orbits that implies strong restrictions on their joinings which in turn yields higher order mixing this is the first case in which the ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications

improved kam estimates for the siegel radius for the siegel center problem we explore the possibility of improving the kam estimates with a view to possible extensions to hamiltonian systems the use of a suitable norm and explicit perturbative computations allow estimates to within a factor 2 of the siegel radius for the quadratic map

classification of all cycles of the parabolic map abstract the problem of determining all cycles of the parabolic map p x for all parameters values 0 and all x 0 is addressed the method is one of decomposing the set of all fixed points of the nth iterate p n x for arbitrary n into cycles of p x toward this goal the inverse graph of p n x is constructed for all and its complete description given in terms of sequences 0 1 2 each i a nonnegative integer and their negatives and a total order relation on all such sequences it is shown how the properties of these sequences give a complete description of the evolution of the graph in the parameter with respect to changes in its shape and in its bifurcation structure these results are then used to infer properties of the decomposition of the set of fixed points of p n into cycles of p

chaos for a damped and forced kdv equation abstract the attractor for a damped and forced korteweg de vries equation is studied for large damping it is shown that the attractor is trivial with all solutions converging to a unique fixed point for small damping however the dynamics can be complicated it is investigated numerically how the attractor changes when the damping term decreases in particular a period doubling cascade of periodic solutions is found culminating with a chaotic regime the ratio between consecutive points in this cascade converging to feigenbaum s constant beyond this cascade the frequency spectrum of the solution is continuous and one lyapunov exponent is positive this seems to be the first work to obtain the onset of temporal chaos for this equation

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

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

a system of alternately excited coupled non autonomous oscillators manifesting phenomena intrinsic to complex analytical maps a feasible model is introduced that manifests phenomena intrinsic to iterative complex analytical maps such as the mandelbrot set and julia sets the system is composed of two alternately excited coupled oscillators the idea is based on a turn by turn transfer of the excitation from one subsystem to another s p kuznetsov example of a physical system with a hyperbolic attractor of the smale williams type phys rev lett 95 2005 144101 accompanied with appropriate non linear transformation of the complex amplitude of the oscillations in the course of the process analytical and numerical studies are performed special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations the main effect is the rotation of the mandelbrot like set in the complex parameter plane one more effect is the destruction of subtle small scale fractal structure of the set due to the presence of non analytical terms in the complex amplitude equations

mandelbrot set in coupled logistic maps and in an electronic experiment we suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps the idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one dimensional quadratic maps with a particular coupling hence dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems which relate to the feigenbaum universality class to substantiate the feasibility of this concept we consider an electronic system which exhibits dynamical phenomena intrinsic to complex analytic maps experimental results are presented providing the mandelbrot set in the parameter plane of this physical system

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

multi modal extended solutions of the feigenbaum equation abstract the feigenbaum equation describes universal features of period doubling in interval maps based on the known result on existence of continuous solutions efforts have been made to find specific classes of continuous solutions which exhibit structure of various continuous solutions in addition to solutions of simpler structure such as bijective solutions some non monotonic solutions such as unimodal solutions and unimodal extended solutions were obtained in this paper we find a new and more complicated class of solutions called multi modal extended solutions and illustrate their structure with 2 modal extended solutions furthermore we give conditions to guarantee smoothness of those solutions

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator