multifractal analysis of conformal axiom a flows we develop the multifractal analysis of conformal axiom a flows this includes the study of the hausdorff dimension of basic sets of the flow the description of the dimension spectra for pointwise dimension and for lyapunov exponents and the multifractal decomposition associated with these spectra the main tool of study is the thermodynamic formalism for hyperbolic flows by bowen and ruelle examples include suspensions over axiom a conformal diffeomorphisms anosov flows and in particular geodesic flows on compact smooth surfaces of negative curvature

critical dynamics and trees abstract we show that the dependence of the critical perturbation of hamiltonian systems on the frequency v is related to the statistics of large planar trees using a complex map for illustration a critical function k v is given by the radii of convergence of power series whose coefficients are represented as sums over trees a preliminary approximation based on this relation leads to an expression for the critical function as the solution of a transcendental equation which appears to provide a reasonable approximation to the true function and to have the right structure near the dense set of singularities

microscopic chaos and gaussian diffusion processes abstract in this paper we construct and analyze a prototypical model of microscopic chaos in particular we extend the results of beck and shimizu to the case where the microscopic time scale is no longer small the upshot is that a non ornstein uhlenbeck deterministic process can generate a gaussian diffusion process

superstatistics in hydrodynamic turbulence abstract superstatistics is a statistics of a statistics relevant for driven nonequilibrium systems with fluctuating intensive parameters it contains tsallis statistics as a special case we show that probability density functions of velocity differences and accelerations measured in eulerian and lagrangian turbulence experiments are well reproduced by simple superstatistics models we compare fits obtained for log normal superstatistics and 2 superstatistics tsallis statistics

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

on the classification of g spheres i equivariant transversality this paper is the first in a series of three stated in geometric terms the papers examine locally linear group actions on spheres for odd order groups g in essentially equivalent homotopy theoretic terms the papers study the homotopy types of the spaces plc v and tope v of equivariant pl homeomorphisms and homeomorphisms of a linear representation v in fact it is the homogeneous spaces f6 v plg v and top6 v pl6 v we study where fc v is the space of proper equivariant homotopy equivalences our results generalize theorems of haefliger kirby siebenmann sullivan and wall and others the link between classification of manifolds and classification of homotopy types is transversality and this is the subject of the present first paper the second paper generalizes wall s classification of fake lens spaces to the classification of g spheres which are equivariantly homotopy equivalent to a given linear action this involves the determination of the equivariant homotopy type of the g space f pl and a complete calculation of the pl equivariant surgery sequence for a linear g sphere as a result we show that the homotopy groups of pl v u plc v vanish in dimensions less than dim v g in the third paper we study g triangulation theory in particular we study the homotopy groups of top6 v u topc v in a range the homotopy groups of the stiefel spaces plc v u plc v and topc v u top6 v are in turn needed for equivariant transversality so the three papers are locked together in an inductive fashion the reader is referred to w 4 below and to the individual parts ii and iii for more information we now give a brief discussion of the content of the present paper let g denote an arbitrary finite group we consider the g transversality question in the locally linear

higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

multifractal analysis of hyperbolic flows we establish the multifractal analysis of hyperbolic flows and of suspension flows over subshifts of finite type a non trivial consequence of our results is that for every holder continuous function non cohomologous to a constant the set of points without birkhoff average has full topological entropy

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

meromorphic zeta functions for analytic flows we extend to hyperbolic flows in all dimensions rugh s results on the meromorphic continuation of dynamical zeta functions in particular we show that the ruelle zeta function of a negatively curved real analytic manifold extends to a meromorphic function on the complex plane

bifurcations of subharmonics abstract we consider a piecewise linear second order ordinary differential equation which is topologically equivalent to the sine gordon equation the system is subjected to a time harmonic disturbance and the behavior of the periodic solutions is examined it is shown that subharmonic motions with period n times that of the disturbance appear via saddle node bifurcations for all n 1 2 3 these motions then undergo period doubling or pitchfork bifurcations as parameters are varied beyond the saddle node bifurcation values the limit n is considered and is compared with a melnikov calculation for the system due to the piecewise linear nature of the system no approximations are necessary and thus the results are valid for nonsmall parameter values furthermore this provides an answer to the uniformity question in the study of stability of subharmonics

approach to gaussian stochastic behavior for systems driven by deterministic chaotic forces we consider skew product dynamical systems that describe the stroboscopic dynamics of a damped particle subjected to a chaotic kick force in a suitable scaling limit the dynamics converges to the ornstein uhlenbeck process we investigate the deterministic chaotic corrections in the vicinity of this gaussian limit case for various examples of chaotic forces we present numerical evidence that for certain classes of chaotic forces the deterministic chaotic corrections of the invariant density are universal we provide analytical results for forces generated by tchebyscheff maps and sketch a renormalization group theory in the space of probability densities

on the fluctuation relation for nose hoover boundary thermostated systems we discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures the system is a modified lorentz gas in which the fixed scatterers exchange energy with the gas of particles and the thermostats are modelled by two nose hoover thermostats applied at the boundaries of the system the transient fluctuation relation which holds only for a precise choice of the initial ensemble is verified at all times as expected times longer than the mesoscopic scale needed for local equilibrium to be settled are required if a different initial ensemble is considered this shows how the transient fluctuation relation asymptotically leads to the steady state relation when as explicitly checked in our systems the condition found in d j searles et al j stat phys 128 1337 2007 for the validity of the steady state fluctuation relation is verified for the steady state fluctuations of the phase space contraction rate and of the dissipation function a similar relaxation regime at shorter averaging times is found the quantity satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale the quantity appears to begin a monotonic convergence after such times this is consistent with the fact that and differ by a total time derivative and that the tails of the probability distribution function of are gaussian

application of generalized thermostatistics to fully developed turbulence abstract we apply the formalism of non extensive statistical mechanics to fully developed turbulent flows we obtain analytical formulas for probability density functions of velocity differences depending on distance r and reynolds number r which are in very good agreement with turbulence experiments the skewness is taken into account by analysing universal contributions due to higher order correlations of the chaotic dynamics we also calculate the moment scaling exponents m which also agree well with experimental data

chaotic resonance two state model with chaos induced escape over potential barrier we consider the resonant effects of chaotic fluctuations on a strongly damped particle in a bistable potential driven by weak sinusoidal perturbation we derive analytical expressions of chaos induced transition rate between the neighboring potential wells based on the inhomogeneous smoluchowski equation our first order analysis reveals that the transition rate has the form of the kramers escape rate except for a perturbed prefactor this modification to the prefactor is found to arise from the statistical asymmetry of the chaotic noise by means of the two state model and the chaos induced transition rate we arrive at an analytical expression of the signal to noise ratio snr our first order snr shows that chaotic resonance can correspond directly to stochastic resonance

alexander duality and hurewicz fibrations we explore conditions under which the restriction of the projection map p sn x b b to an open subset u c sn x b is a hurewicz fibration as a consequence we exhibit hurewicz fibrations p e y i such that i p e i is not a locally trivial bundle ii p 1 t is an open n manifold for each t and iii p o proj e x r1 i is a locally trivial bundle the fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity we also show that when the fibers of a hurewicz fibration with open n manifold fibers have finitely generated n l st homology then all fibers have the same finite number of ends this last shows that the punctured torus and the thrice punctured two sphere cannot both be fibers of a hurewicz fibration p e i with open 2 manifold fibers

relaxation and bifurcation in brownian motion driven by a chaotic force brownian motion driven by a chaotic sequence of iterates of a map f y which may depend on a bifurcation parameter is discussed n t t t where t kyn 1 for n t n 1 n 0 1 2 and yn 1 f yn the time evolution equation for the distribution function of the velocity is derived if 1 the distribution tends to the stationary one which has a similar form as the invariant density of f y in the limit of 0 the distribution function satisfies a fokker planck type equation with memory effects the dependence of relaxation processes on the bifurcation parameter is also investigated the fluctuation dissipation theorem is discussed the theoretical results are shown to be in a good agreement with numerical ones which have been done for the tent map and the logistic map

nonuniform measure rigidity we consider an ergodic invariant measure for a smooth actionof z k k 2 on a k 1 dimensional manifold or for a locally free smooth action of r k k 2 on a 2k 1 dimensional manifold we prove that if is hyperbolic with the lyapunov hyperplanes in general position and if one element in z k has positive entropy then is absolutely continuous the main ingredient is absolute continuity of conditional measures on lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups

chaotic cascade model for turbulent velocity distributions a coupled map lattice is introduced that simulates the time evolution of velocity differences in fully developed turbulent flows the model considered is an extension of the langevin theory to chaotic driving forces acting on a self similar cascade of spatial levels compared to full simulations of the navier stokes equation the amount of necessary computing time is negligible despite its simplicity the model is in perfect agreement with experimentally observed results provided the chaotic driving force is generated by the fully developed logistic map with parameter value ensuremath mu 2 the shape of the velocity distributions the slight asymmetry the stretched exponential tails as well as the moment scaling exponents mathrm ensuremath zeta mathit m come out in precisely the same way as in experimental measurements of high reynolds number flows

the differential zeta function for axiom a attractors closed orbit of least period x t an important problem is to construct the meromorphic extensions for these zeta functions a detailed knowledge of the domain of s gives information on the long term behaviour of the flow for example this is a standard approach to deriving asymptotic formulae for closed orbits cf 9 for the case of geodesic flows on surfaces of constant negative curvature the zeta function is known to have a meromorphic extension to the entire complex plane unfortunately an obstruction to a general theory along these lines is the existence of examples of axiom a flows for which d s does not have a meromorphic extension to c in a recent article william parry proposed a modified definition of a zeta function for axiom a attractors he defined the differential zeta function by ns s h i e s t l where the least period x x of a closed orbit is replaced by the expansion xu t along the unstable manifold around the orbit 8 parry also defined the useful notion of a synchronized flow i e one for which the measure of maximal entropy and the srb measure coincide the following remark is taken from page 273 of his article when the flow is synchronized the zeta functions coincide this is the case for example with a geodesic flow on a surface of constant negative curvature one wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow in this article we shall give a solution to parry s question which shows that in many cases du s has a meromorphic extension to the entire complex plane our main result can be stated as follows

closing the hierarchy of moment equations in nonlinear dynamical systems the moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems in the present paper a systematic approach for closing this hierarchy is proposed based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale the method is applied to a one dimensional vector field in the presence of noise and to two prototypes of chaotic behavior excellent agreement with numerical results is obtained special emphasis is placed on the role of symmetries and on the origin of the composite oscillations found for certain types of moments in the chaotic systems

an invariance principle for maps with polynomial decay of correlations we give a general method of deriving statistical limit theorems such as the central limit theorem and its functional version in the setting of ergodic measure preserving transformations this method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations

multifractal analysis of nonhyperbolic coupled map lattices application to genomic sequences symbolic sequences generated by coupled map lattices cmls can be used to model the chaotic like structure of genomic sequences in this study it is shown that diffusively coupled chebyshev maps of order 4 corresponding to a shift of four symbols very closely reproduce the multifractal spectrum d q of human genomic sequences for coupling constant 0 35 0 01 if q 0 the presence of rare configurations causes deviations for q 0 which disappear if the rare event statistics of the cml is modified such rare configurations are known to play specific functional roles in genomic sequences serving as promoters or regulatory elements

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

equivalence of nonequilibrium ensembles for simple maps abstract in this paper we discuss the equivalence of different approaches to the same transport problems with particular attention to recent works by breyman et al these suggest a singular equivalence between the stationary states of models from gaussian dynamics and those from chaotic scattering theory although the mentioned equivalence may concern but a limited set of the properties of those systems it is very important from a theoretical standpoint we argue that a special form of dissipativity in the evolution equations of the phase space distributions is to be invoked to understand those facts we discuss one application of periodic orbit theory to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained

the number of periodic solutions of 2 dimensional periodic systems on donne des conditions suffisantes sur un systeme a 2 dimensions periodique qui a un nombre fini de solutions periodiques

impact oscillators with homoclinic orbit tangent to the wall abstract homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall consequently the impact surface cannot be chosen as the poincare section to measure the distance between the perturbed stable and unstable manifolds furthermore compared to the case that the unperturbed homoclinic orbit intersects the wall transversally more cases are involved as the parameters vary thus the analysis of the homoclinic orbit tangent to the wall is much more difficult in this paper by using a method of melnikov type we derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally as an application an impact oscillator of duffing type is studied in detail

synchronisation of canonical measures for hyperbolic attractors under suitable conditions it is shown how to change the velocity of ac2 axioma attractor so that the sinai ruelle bowen measure coincides with the measure of maximal entropy these measures are obtained as limits of certain closed orbital measures

ricci curvature and volume convergence the purpose of this paper is to give a new integral estimate of distances and angles on manifolds with a given lower ricci curvature bound this will provide us with an integral version of the toponogov comparison triangle theorem for ricci curvature and almost extreme triangles see the earlier works cl and c2 for an analog of this when the manifold has positive ricci curvature using this we prove the anderson cheeger conjecture saying that the volume is a continuous function on the space of all closed n manifolds with ricci curvature greater or equal to n 1 equipped with the gromovhausdorff metric we also prove gromov s conjecture for n 57 3 saying that an almost nonnegatively ricci curved n manifold with first betti number equal to n is a torus further we prove a conjecture of anderson cheeger saying that an open n manifold with nonnegative ricci curvature whose tangent cone at infinity is in is itself in finally we prove a conjecture of fukaya yamaguchi we will now describe these results in more detail let dgh denote the gromov hausdorff distance glp first we have the following result which was conjectured by anderson cheeger

submersions fibrations and bundles when does a submersion have the homotopy lifting property when is it a locally trivial fibre bundle we establish characterizations in terms of consistency in the topology of the neighbouring fibres

from ruelle s transfer operator to the schro dinger operator abstract we show that ruelle s transfer operator defined for an appropriate class of strongly chaotic mappings reduces to the schrodinger operator with imaginary time in an appropriate scaling limit we also derive the equations for the next order corrections to the schrodinger behavior if the scaling limit is not completely performed in the simplest case the first order corrections turns out to obey a schrodinger equation with a source term this means a possible chaotic dynamics underlying the feynman kac formula on a very small time scale e g the planc scale leads generalized quantum mechanics with an additional source term

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

the steady state fluctuation relation for the dissipation function we give a proof of transient fluctuation relations for the entropy production dissipation function in nonequilibrium systems which is valid for most time reversible dynamics we then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax producing a unique nonequilibrium steady state although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown if such a steady state exists the generation of the steady state fluctuation relation from the transient relation is shown to be very general it is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation which is needed also for e g the existence of transport coefficients because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities the steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor whose convergence is problematic for systems close to equilibrium we examine some model dynamics that have been considered previously and show how they are described in the context of this work

from exactly solvable chaotic maps to stochastic dynamics abstract for a class of nonlinear chaotic maps the exact solution can be written as xn p kn where p t is a periodic function is a real parameter and k is an integer number a generalization of these functions xn p zn where z is a real parameter can be proved to produce truly random sequences using different functions p t we can obtain different distributions for the random sequences similar results can be obtained with functions of type xn h f n where f n is a chaotic function and h t is a noninvertible function we show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics we present physical systems with this kind of behavior we report the results of real experiments with nonlinear circuits and josephson junctions we show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems we discuss applications of these phenomena in dynamics based computation

systems of fixed point sets let g be a compact lie group a canonical method is given for constructing a c space from homotopy theoretic information about its fixed point sets the construction is a special case of the categorical bar construction applications include easy constructions of certain classifying spaces as well as c eilenbergmac lane spaces and postnikov towers 0 introduction let g be a compact lie group and x a g space the equivariant homotopy theory of x is reflected to a remarkable extent in its system of fixed point sets defined as a functor from a certain category 0g to top the category of topological spaces our spaces will be compactly generated weak hausdorff they may or may not be equipped with a basepoint depending on the context these functors or systems have considerable technical advantages over g spaces it is easy to apply most homotopy theoretic constructions to them whereas in many cases it is unclear how to proceed for g spaces it is the purpose of this paper to present a canonical way of recovering from any system a g space which preserves all the homotopy theoretic structure of the system this allows us to give easy equivariant versions of some standard topological constructions such as eilenberg mac lane spaces and postnikov towers and to simplify other equivariant constructions 1 1 statements of the main theorems throughout g is a fixed compact lie group definitions the category of canonical orbits written 0g is a topological category with discrete object space 0g g 77 77 a closed subgroup of g and morphisms the g maps topologized by requiring the natural bijection hoxn0cig h g k g k h to be a homeomorphism by an oc space we shall mean a continuous contravariant functor from 0g to top these functors form the objects of a topological category in the usual manner we will also consider gg rings oc groups etc defined similarly definition let a1 be a g space the fixed point set system of x written x is an gg space defined as follows g 77 x received by the editors october 30 1981 and in revised form april 26 1982 1980 mathematics subject classification primary 57s10 55n25

stable ergodic properties of cocycles over hyperbolic attractors we consider a hyperbolic flow t defined on an attracting basic set a map from the first cech cohomology group of into the dynamic cohomology group is constructed this map is used to discuss the stable ergodicity and mixing of compact lie group extensions and velocity changes of t

oseledets regularity functions for anosov flows oseledets regularity functions quantify the deviation of the growth associated with a dynamical system along its lyapunov bundles from the corresponding uniform exponential growth the precise degree of regularity of these functions is unknown we show that for every invariant lyapunov bundle of a volume preserving anosov flow on a closed smooth riemannian manifold the corresponding oseledets regularity functions are in l p m for some p 0 where m is the probability measure defined by the volume form we prove an analogous result for essentially bounded cocycles over volume preserving anosov flows

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3