periodic shadowing sensitivity analysis of chaotic systems abstract the sensitivity of long time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction the uniformly bounded in time solution of the sensitivity equations although its existence is formally guaranteed for certain systems methods to determine it are hardly available one practical approach is the least squares shadowing lss algorithm wang 2014 18 whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm here we present an alternative potentially simpler shadowing based algorithm termed periodic shadowing the key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time we show that this is not only justifiable when the reference trajectory is itself periodic but also possible and effective for chaotic trajectories our error analysis shows that periodic shadowing has the same convergence rates as lss when the time span t is increased the sensitivity error first decays as 1 t and then asymptotically as 1 t we demonstrate the approach on the lorenz equations and also show that as t tends to infinity periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits lasagna 2018 14 for which there is no shadowing error finally finite difference approximations of the sensitivity are also examined and we show that subtle non hyperbolicity features of the lorenz system introduce a small yet systematic bias

on spurious detection of linear response and misuse of the fluctuation dissipation theorem in finite time series abstract using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems a goodness of fit statistics is developed for a linear statistical model of the observations based on results for central limit theorems for deterministic dynamical systems and used to detect linear response breakdown we apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series the magnitude of the perturbation and on the choice of the observable we find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed even for simple systems such as the logistic map one needs of the order of 10 6 observations to reliably detect the breakdown with a confidence level of 95 if less observations are available one may be falsely led to conclude that linear response theory is valid the amount of data required is larger the smaller the applied perturbation for judiciously chosen observables the necessary amount of data can be drastically reduced but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems furthermore we explore the use of the fluctuation dissipation theorem fdt in cases with limited data length or coarse graining of observations the fdt if applied naively to a system without linear response is shown to be very sensitive to the details of the sampling method resulting in erroneous predictions of the response

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

higher dimensional algebra i braided monoidal 2 categories abstract we begin with a brief sketch of what is known and conjectured concerning braided monoidal 2 categories and their relevance to 4d tqfts and 2 tangles then we give concise definitions of semistrict monoidal 2 categories and braided monoidal 2 categories and show how these may be unpacked to give long explicit definitions similar to but not quite the same as those given by previously kapranov and voevodsky finally we describe how to construct a semistrict braided monoidal 2 category z c as the center of a semistrict monoidal category c in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category as a corollary this yields a strictification theorem for braided monoidal 2 categories

evidence of dispersion relations for the nonlinear response of the lorenz 63 system along the lines of the nonlinear response theory developed by ruelle in a previous paper we have proved under rather general conditions that kramers kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of axiom a non equilibrium steady state systems to weak monochromatic forcings we present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed lorenz 63 system by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions some new theoretical results showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables are also presented our findings confirm the conceptual validity of the nonlinear response theory suggest that the theory can be extended for more general non equilibrium steady state systems and shed new light on the applicability of very general tools based only upon the principle of causality for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals in situations where the fluctuation dissipation relation is not of great help

stochastic climate dynamics random attractors and time dependent invariant measures a b s t r a c t this article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s the nonlinear deterministic and the linear stochastic one this unification via the theory of random dynamical systems rds allows one to consider the detailed geometric structure of the random attractors associated with nonlinear stochastically perturbed systems we report on high resolution numerical studies of two idealized models of fundamental interest for climate dynamics the first of the two is a stochastically forced version of the classical lorenz model the second one is a low dimensional nonlinear stochastic model of the el nino southern oscillation enso these studies provide a good approximation of the two models global random attractors as well as of the time dependent invariant measures supported by these attractors the latter are shown to have an intuitive physical interpretation as random versions of sinai ruelle bowen srb measures

spectral galerkin methods for transfer operators in uniformly expanding dynamics markov expanding maps a class of simple chaotic systems are commonly used as models for chaotic dynamics but existing numerical methods to study long time statistical properties such as invariant measures have a poor trade off between computational effort and accuracy we develop a spectral galerkin method for these maps transfer operators estimating statistical quantities using finite submatrices of the transfer operators infinite fourier or chebyshev basis coefficient matrices rates of convergence of these estimates are obtained via quantitative bounds on the full transfer operator matrix entries we find the method furnishes up to exponentially accurate estimates of statistical properties in only a polynomially large computational time to implement these results we suggest and demonstrate two algorithms a rigorously validated algorithm and a fast more convenient adaptive algorithm using the first algorithm we prove rigorous bounds on some exemplar quantities that are substantially more accurate than previous we show that the adaptive algorithm can produce double floating point accuracy estimates in a fraction of a second on a personal computer

lattice topological field theory in two dimensions the lattice definition of a two dimensional topological field theory tft is given generically and the exact solution is obtained explicitly in particular the set of all lattice topological field theories is shown to be in one to one correspondence with the set of all associative algebrasr and the physical hilbert space is identified with the centerz r of the associative algebrar perturbations of tft s are also considered in this approach showing that the form of topological perturbations is automatically determined and that all tft s are obtained from one tft by such perturbations several examples are presented including twistedn 2 minimal topological matter and the case wherer is a group ring

differentiation of srb states correction and complements taking into account criticism by d dolgopyat and m jiang we present here an improved derivation of the formula for the derivative of an srb measure with respect to parameters

topological higher gauge theory from bf to bfcg theory we study generalizations of three and four dimensional bf theories in the context of higher gauge theory first we construct topological higher gauge theories as discrete state sum models and explain how they are related to the state sums of yetter mackaay and porter under certain conditions we can present their corresponding continuum counterparts in terms of classical lagrangians we then explain that two of these models are already familiar from the literature the ea model of three dimensional gravity coupled to topological matter and also a four dimensional model of bf theory coupled to topological matter

monoidal globular categories as a natural environment for the theory of weakn categories we present a definition of weak categories based on a higher order generalization of apparatus of operads

response theory for equilibrium and non equilibrium statistical mechanics causality and generalized kramers kronig relations we consider the general response theory recently proposed by ruelle for describing the impact of small perturbations to the non equilibrium steady states resulting from axiom a dynamical systems we show that the causality of the response functions entails the possibility of writing a set of kramers kronig k k relations for the corresponding susceptibilities at all orders of nonlinearity nonetheless only a special class of directly observable susceptibilities obey k k relations specific results are provided for the case of arbitrary order harmonic response which allows for a very comprehensive k k analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short time response of the perturbed system these results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models and shed light on the very general impact of considering the principle of causality for testing self consistency the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey the theory exposed in the present paper is dual to the time dependent theory of perturbations to equilibrium states and to non equilibrium steady states and has in principle similar range of applicability and limitations in order to connect the equilibrium and the non equilibrium steady state case we show how to rewrite the classical response theory by kubo so that response functions formally identical to those proposed by ruelle apart from the measure involved in the phase space integration are obtained these results taking into account the chaotic hypothesis by gallavotti and cohen might be relevant in several fields including climate research in particular whereas the fluctuation dissipation theorem does not work for non equilibrium systems because of the non equivalence between internal and external fluctuations k k relations might be robust tools for the definition of a self consistent theory of climate change

stochastic perturbations to dynamical systems a response theory approach using the formalism of the ruelle response theory we study how the invariant measure of an axiom a dynamical system changes as a result of adding noise and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics we first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system both in the additive and in the multiplicative case we also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing this provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings using kramers kronig theory it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the green function of the system for any desired observable we then extend our results to rather general patterns of random forcing from the case of several white noise forcings to noise terms with memory up to the case of a space time random field explicit formulas are provided for each relevant case analysed as a general result we find using an argument of positive definiteness that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system we provide an example of application of our results by considering the spatially extended chaotic lorenz 96 model these results clarify the property of stochastic stability of srb measures in axiom a flows provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies and introduce new ways to study the response of a system to external perturbations taking into account the chaotic hypothesis we expect that our results have practical relevance for a more general class of system than those belonging to axiom a

response operators for markov processes in a finite state space radius of convergence and link to the response theory for axiom a systems using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state markov processes the results can be used for studying empirically constructed e g from observations or through coarse graining of model simulations finite state approximation of statistical mechanical systems recent results concerning the convergence of the statistical properties of finite state markov approximation of the full asymptotic dynamics on the srb measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of axiom a system our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language plus providing bounds on the radius of convergence of the perturbative theory in particular we relate the convergence of the response theory to the rate of mixing of the unperturbed system one can use the formulas derived for finite state markov processes to recover previous findings obtained on the response of continuous time axiom a dynamical systems to perturbations by considering the generator of time evolution for the measure and for the observables a very basic low tech and computationally cheap analysis of the response of the lorenz 63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state markov processes to compute the system s response

state sum models and simplicial cohomology we study a class of subdivision invariant lattice models based on the gauge groupz p with particular emphasis on the four dimensional example this model is based upon the assignment of field variables to both the 1 and 2 dimensional simplices of the simplicial complex the property of subdivision invariance is achieved when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod p flatness condition by explicit computation of the partition function for the manifoldrp3 s1 we establish that the theory has a quantum hilbert space which differs from the classical one

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

application of hyperbolic dynamics to physics some problems and conjectures the long time behavior of smooth dynamical systems is in good cases given by an srb measure the measure is expected often to have very discontinuous dependence on parameters of the dynamical system this is a very unsatisfactory situation for physical applications where one would like to differentiate with respect to parameters here we propose a solution to this difficulty based on analytic continuation in a frequency variable rene thom was a philosopher and among other things but most successfully a lover of that part of philosophy which we call mathematics i use here the term philosophy with its old meaning which some contemporary philosophers and mathematicians might not like inside and outside mathematics thom s interest was turned to forms and ideas his passion was to understand geometrically the nature of things and for this he used mathematical proofs but he was not a formalist and for him proofs remained secondary to the conceptual landscape that they revealed one area where he invested a lot of effort was the study of differentiable dynamical systems this smooth dynamics is where geometric forms come closest to life i attended the regular seminar that he ran on the subject under the influence of thom the ihes was an important center for the development of differentiable dynamical systems especially hyperbolic dynamics in the golden years of the 1960 s and 1970 s steve smale and his friends were frequent visitors and the russians came when they could the big question in smooth dynamics is what do general smooth dynamical systems look like what one wants is an understanding of a large set of dynamical systems this is also a significant question for physical applications more specifically one would like to understand the long time behavior for typical time evolutions and how this changes when the parameters controlling the time evolution are changed the uniformly hyperbolic systems studied by anosov smale thom and many others form a large set and smale had conjectured that they might be dense among all dynamical systems but this is not the case and it may be too much to hope for a very large set of dynamical systems that are understood in great detail like the uniformly hyperbolic systems from a practical point of view the numerical study of dynamical systems often reveals extremely long characteristic times which received by the editors october 1 2003 2000 mathematics subject classification primary 37c40 secondary 37d45

towards a general theory of extremes for observables of chaotic dynamical systems in this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values we show that the extremes of so called physical observables are distributed according to the classical generalised pareto distribution and derive explicit expressions for the scaling and the shape parameter in particular we derive that the shape parameter does not depend on the chosen observables but only on the partial dimensions of the invariant measure on the stable unstable and neutral manifolds the shape parameter is negative and is close to zero when high dimensional systems are considered this result agrees with what was derived recently using the generalized extreme value approach combining the results obtained using such physical observables and the properties of the extremes of distance observables it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow moreover by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments of the partial dimensions and of the kaplan yorke dimension of the attractor preliminary numerical investigations provide encouraging results on the applicability of the theory presented here the results presented here do not apply for all combinations of axiom a systems and observables but the breakdown seems to be related to very special geometrical configurations

whitney holder continuity of the srb measure for transversal families of smooth unimodal maps we consider c 2 families t mapsto f t of c 4 nondegenerate unimodal maps we study the absolutely continuous invariant probability srb measure mu t of f t as a function of t on the set of collet eckmann ce parameters upper bounds assuming existence of a transversal ce parameter we find a positive measure set of ce parameters delta and for each t 0 in delta a set delta 0 subset delta of polynomially recurrent parameters containing t 0 as a lebesgue density point and constants c ge 1 gamma 4 so that for every 1 2 holder function a begin aligned big int a d mu t int a d mu t 0 big le c vert a vert c 1 2 t t 0 1 2 log t t 0 gamma forall t in delta 0 end aligned in addition for all t in delta 0 the renormalisation period p t of f t satisfies p t le p t 0 and there are uniform bounds on the rates of mixing of f t p t for all t with p t p t 0 if f t x tx 1 x the set delta contains almost all ce parameters lower bounds assuming existence of a transversal mixing misiurewicz thurston parameter t 0 we find a set of ce parameters delta mt accumulating at t 0 a constant c ge 1 and a c infty function a 0 so that begin aligned c t t 0 1 2 ge big int a 0 d mu t int a 0 d mu t 0 big ge c 1 t t 0 1 2 forall t in delta mt end aligned

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

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

subspaces generated by translations in rearrangement invariant spaces in the setting of rearrangement invariant r i banach function spaces e on 0 we study the complementability of subspaces qa generated by sequences of translations of functions a e 0 1 an r i function space e is said to be nice in short e n if every subspace of type qa is complemented we give necessary and sufficient conditions for an r i function space to be nice we determinate the orlicz lorentz and marcinkiewicz spaces belonging to the class n as an application we obtain a new characterization of the lp spaces 1 p among the class of r i function spaces

linear response function for coupled hyperbolic attractors the srb measures of a hyperbolic system are widely accepted as the measures that are physically relevant it has been shown by ruelle that they depend smoothly on the system furthermore ruelle showed by a separate argument that the first derivative i e the linear response function admits a geometric interpretation in this paper we consider thermodynamic limits of srb measures in lattices of coupled hyperbolic attractors in a previous paper using markov partitions and thermodynamic formalism we had established the smooth dependence of thermodynamic limits of srb measures here we establish that the linear response function admits a geometric interpretation the formula is analogous to the one found by ruelle for finite dimensional systems if one term is reinterpreted appropriately we show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume we also obtain similar results for the derivatives of the entropy

finite groups spherical 2 categories and 4 manifold invariants in this paper we define a class of state sum invariants of closed oriented piece wise linear 4 manifolds using finite groups the defini tion of these state sums follows from the general abstract construction of 4 manifold invariants using spherical 2 categories as we defined in an earlier paper we show that the state sum invariants of birming ham and rakowski who studied dijkgraaf witten type invariants in dimension 4 are special examples of the general construction that we present in this paper they showed that their invariants are non trivial by some explicit computations so our construction includes interest ing examples already finally we indicate how our construction is related to homotopy 3 types this connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3 types by brown for example

generalized centers of braided and sylleptic monoidal 2 categories abstract recent developments in higher dimensional algebra due to kapranov and voevodsky day and street and baez and neuchl include definitions of braided sylleptic and symmetric monoidal 2 categories and a center construction for monoidal 2 categories which gives a braided monoidal 2 category i give generalized center constructions for braided and sylleptic monoidal 2 categories which give sylleptic and symmetric monoidal 2 categories respectively and i correct some errors in the original center construction for monoidal 2 categories

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

higher dimensional algebra iv 2 tangles just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions compact surfaces smoothly embedded in r4 can be described as certain 2 morphisms in the 2 category of 2 tangles in 4 dimensions using the work of carter rieger and saito we prove that this 2 category is the free semistrict braided monoidal 2 category with duals on one unframed self dual object by this universal property any unframed self dual object in a braided monoidal 2 category with duals determines an invariant of 2 tangles in 4 dimensions

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

sensitivity of long periodic orbits of chaotic systems the properties of long numerically determined periodic orbits of two low dimensional chaotic systems the lorenz equations and the kuramoto sivashinsky system in a minimal domain configuration are examined the primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite amplitude parameter perturbations to address this question an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed the expectation of period averages floquet exponents and sensitivities over such set is then obtained it is shown that all these quantities converge to a limiting value as the orbit period is increased however while period averages and floquet exponents appear to converge to analogous quantities computed from chaotic trajectories the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state similar to observations made with other shadowing algorithms

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

holonomy and parallel transport for abelian gerbes in this paper we establish a one to one correspondence between u 1 gerbes with connections on the one hand and their holonomies for simply connected manifolds or their parallel transports in the general case on the other hand this result is a higher order analogue of the familiar equivalence between bundles with connections and their holonomies for connected manifolds the holonomy of a gerbe with group u 1 on a simply connected manifold m is a group morphism from the thin second homotopy group to u 1 satisfying a smoothness condition where a homotopy between maps from 0 1 2 to m is thin when its derivative is of rank 2 for the non simply connected case holonomy is replaced by a parallel transport functor between two special lie groupoids which we call lie 2 groups the reconstruction of the gerbe and connection from its holonomy is carried out in detail for the simply connected case

multiplicator space and complemented subspaces of rearrangement invariant space abstract we show that the multiplicator space m x of an rearrangement invariant r i space x on 0 1 and the nice part n 0 x of x that is the set of all a x for which the subspaces generated by sequences of dilations and translations of a are uniformly complemented coincide when the space x is separable in the general case the nice part is larger than the multiplicator space several examples of descriptions of m x and n 0 x for concrete x are presented

the natural rearrangement invariant structure on tensor products we prove that the only rearrangement invariant r i spaces for which there exists a crossnorm verifying that the tensor product of these spaces preserves the natural r i space structure in the sense that it makes the multiplication operator b a topological isomorphism are the lp spaces

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

structure and f dependence of the a c i m for a unimodal map f of misiurewicz type by using a suitable banach space on which we let the transfer operator act we make a detailed study of the ergodic theory of a unimodal map f of the interval in the misiurewicz case we show in particular that the absolutely continuous invariant measure can be written as the sum of 1 square root spikes along the critical orbit plus a continuous background we conclude by a discussion of the sense in which the map f mapsto rho may be differentiable

statistical properties of unimodal maps the quadratic family we prove that almost every nonregular real quadratic map is colleteckmann and has polynomial recurrence of the critical orbit proving a conjecture by sinai it follows that typical quadratic maps have excellent ergodic properties as exponential decay of correlations keller and nowicki young and stochastic stability in the strong sense baladi and viana this is an important step in achieving the same results for more general families of unimodal maps

cohomology and deformation theory of monoidal 2 categories i abstract in this paper we define a cohomology theory for an arbitrary k linear semistrict semigroupal 2 category c called for short a gray semigroup and show that its first order unitary deformations up to the suitable notion of equivalence are in bijection with the elements of the second cohomology group fundamental to the construction is a double complex similar to the gerstenhaber schack double complex for bialgebras the role of the multiplication and the comultiplication being now played by the composition and the tensor product of 1 morphisms we also identify the cohomologies describing separately the deformations of the tensor product the associator and the pentagonator to obtain the above results a cohomology theory for an arbitrary k linear unitary pseudofunctor is introduced describing its purely pseudofunctorial deformations and generalizing yetter s cohomology for semigroupal functors in m kapranov e getzler eds higher category theory ams contemporary mathematics vol 230 amer math soc providence ri 1998 pp 117 134 the corresponding higher order obstructions will be considered in detail in a future paper

structures and diagrammatics of four dimensional topological lattice field theories abstract crane and frenkel proposed a state sum invariant for triangulated 4 manifolds they sketched the definition of a hopf category that was to be used in their construction crane and yetter studied hopf categories and gave some examples using group cocycles that are associated to the drinfeld double of a finite group in this paper we define a state sum invariant of triangulated 4 manifolds using crane yetter cocycles as boltzmann weights our invariant is analogous to the 3 dimensional invariants defined by dijkgraaf and witten and the invariants that are defined via hopf algebras we present diagrammatic methods for the study of such invariants that illustrate connections between hopf categories and moves to triangulations

linear response for intermittent maps we consider the one parameter family alpha mapsto t alpha alpha in 0 1 of pomeau manneville type interval maps t alpha x x 1 2 alpha x alpha for x in 0 1 2 and t alpha x 2x 1 for x in 1 2 1 with the associated absolutely continuous invariant probability measure mu alpha for alpha in 0 1 sarig and gouezel proved that the system mixes only polynomially with rate n 1 1 alpha in particular there is no spectral gap we show that for any psi in l q the map alpha to int 0 1 psi d mu alpha is differentiable on 0 1 1 q and we give a linear response formula for the value of the derivative this is the first time that a linear response formula for the srb measure is obtained in the setting of slowly mixing dynamics our argument shows how cone techniques can be used in this context for alpha ge 1 2 we need the n 1 alpha decorrelation obtained by gouezel under additional conditions

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

on the validity of linear response theory in high dimensional deterministic dynamical systems this theoretical work considers the following conundrum linear response theory is successfully used by scientists in numerous fields but mathematicians have shown that typical low dimensional dynamical systems violate the theory s assumptions here we provide a proof of concept for the validity of linear response theory in high dimensional deterministic systems for large scale observables we introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large inhomogeneous collection of unresolved chaotic degrees of freedom by employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response we corroborate our result with numerical simulations

higher dimensional algebra ii 2 hilbert spaces abstract a 2 hilbert space is a category with structures and properties analogous to those of a hilbert space more precisely we define a 2 hilbert space to be an abelian category enriched over hilb with a structure conjugate linear on the hom sets satisfying fg h g f h f hg we also define monoidal braided monoidal and symmetric monoidal versions of 2 hilbert spaces which we call 2 h algbebras braided 2 h algebras and symmetric 2 h algbebras and we describe the relation between these and tangles in two three and four dimensions respectively we prove a generalized doplicher roberts theorem starting that every symmetric 2 h algebra is equivalent to the category rep g of continuous unitary finite dimensional representations of some compact supergroupoid g the equivalence is given by a categorified version of the gelfand transform we also construct a categorified version of the fourier transform when g is a compact abelian group finally we characterize rep g by its universal properties when g is a compact classical group for example rep u n is the free connected symmetric 2 h algebra on one even object of dimension n

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

parametric smoothness and self scaling of the statistical properties of a minimal climate model what beyond the mean field theories abstract a quasi geostrophic intermediate complexity model of the mid latitude atmospheric circulation is considered featuring simplified baroclinic conversion and barotropic convergence processes the model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator to pole temperature difference t e as t e increases a transition takes place from a stationary regime hadley equilibrium to a periodic regime and eventually to a chaotic regime where evolution takes place on a strange attractor the attractor dimension metric entropy and bounding box volume in phase space have a smooth dependence on t e which results in power law scaling properties power law scalings with respect to t e are detected also for the statistical properties of global physical observables the total energy of the system and the averaged zonal wind the scaling laws which constitute the main novel result of the present work can be thought to result from the presence of a statistical process of baroclinic adjustment which tends to decrease the equator to pole temperature difference and determines the properties of the attractor of the system the self similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding on a heuristic basis both data analysis and realistic simulations going beyond the unsatisfactory mean field theories and brute force approaches a leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters

a 2 categorical state sum model it has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4 manifolds this idea has been refined recently by proposing to use 2 groups and their representations as specific examples of 2 categories the challenge has been to make these proposals fully explicit here we give a concrete realization of this program building upon our earlier work with baez and wise on the representation theory of 2 groups we construct a four dimensional state sum model based on a categorified version of the euclidean group we define and explicitly compute the simplex weights which may be viewed a categorified analogue of racah wigner 6j symbols these weights solve a hexagon equation that encodes the formal invariance of the state sum under the pachner moves of the triangulation this result unravels the combinatorial formulation of the feynman amplitudes of quantum field theory on flat spacetime proposed in a baratin and l freidel classical quantum gravity 24 2027 2060 2007 which was shown to lead after gauge fixing to korepanov s invariant of 4 manifolds

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

dichromatic state sum models for four manifolds from pivotal functors a family of invariants of smooth oriented four dimensional manifolds is defined via handle decompositions and the kirby calculus of framed link diagrams the invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category a state sum formula for the invariant is constructed via the chain mail procedure so a large class of topological state sum models can be expressed as link invariants most prominently the crane yetter state sum over an arbitrary ribbon fusion category is recovered including the nonmodular case it is shown that the crane yetter invariant for nonmodular categories is stronger than signature and euler invariant a special case is the four dimensional untwisted dijkgraaf witten model derivations of state space dimensions of tqfts arising from the state sum model agree with recent calculations of ground state degeneracies in walker wang models relations to different approaches to quantum gravity such as cartan geometry and teleparallel gravity are also discussed

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

higher dimensional algebra iii n categories and the algebra of opetopes abstract we give a definition of weak n categories based on the theory of operads we work with operads having an arbitrary set s of types or s operads and given such an operad o we denote its set of operations by elt o then for any s operad o there is an elt o operad o whose algebras are s operads over o letting i be the initial operad with a one element set of types and defining i 0 i i i 1 i i we call the operations of i n 1 the n dimensional opetopes opetopes form a category and presheaves on this category are called opetopic sets a weak n category is defined as an opetopic set with certain properties in a manner reminiscent of street s simplicial approach to weak categories similarly starting from an arbitrary operad o instead of i we define n coherent o algebras which are n times categorified analogs of algebras of o examples include monoidal n categories stable n categories virtual n functors and representable n prestacks we also describe how n coherent o algebra objects may be defined in any n 1 coherent o algebra

response formulae for n point correlations in statistical mechanical systems and application to a problem of coarse graining predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences under suitable conditions on the nature of the system of the perturbation and of the observables of interest response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation in particular response theories can be developed both for stochastic and chaotic deterministic dynamical systems where in the latter case stricter conditions imposing some degree of structural stability are required in this paper we extend previous findings and derive general response formulae describing how n point correlations are affected by perturbations to the vector flow we also show how to compute the response of the spectral properties of the system to perturbations we then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system all the terms envisioned by the mori zwanzig theory the deterministic stochastic and non markovian terms are affected at 1st order in the perturbation the obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics condensed matter and geophysical fluid dynamics we envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems