nonparametric estimation of lifetime and disease onset distributions from incomplete observations in this paper we derive and investigate nonparametric estimators of the distributions of lifetime and time until onset associated with an irreversible disease that is detectable only at death the nonparametric maximum likelihood solution requires an iterative algorithm an alternative though closely related pair of estimators for the lifetime and onset distributions exists in closed form these estimators are the familiar kaplan meier estimator and an isotonic regression estimator respectively first order approximations provide variance estimators the proposed methods generalize and shed additional light on the constrained estimators presented by kodell shaw and johnson 1982 biometrics 38 43 58 data from an animal experiment illustrate the techniques

the heat flow with a critical exponential nonlinearity abstract we analyze the possible concentration behavior of heat flows related to the moser trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation as an application of our results we obtain the existence of critical points of the moser trudinger energy in a supercritical regime

a variety of period doubling universality classes in multi parameter analysis of transition to chaos abstract in multi parameter analysis of the onset of chaos non feigenbaum period doubling behavior may occur at some special paths in the parameter space there are two possibilities i the dynamics at the onset of chaos remains essentially one dimensional but the one dimensional map is distorted in such a way that leaves feigenbaum s universality class ii a new mode comes to the threshold of instability and increases the effective dimension of the dynamics we submit a list of one dimensional and two dimensional maps which represent distinct classes of the period doubling universality discuss the properties of the associated types of critical behavior at the border of chaos and demonstrate pictures of the universal parameter space arrangement near the critical points

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

nonparametric methods for survival sacrifice experiments in many carcinogenicity studies the time to disease occurrence is not clinically observable a survival sacrifice experiment is considered for nonparametric inference about the rate of disease occurrence a multistate model for disease development and death is considered and an algorithm of the em type for maximum likelihood estimation is obtained questions of identifiability and estimability are addressed under the model interval hazards for disease occurrence are identifiable for intervals defined by the sacrifice times a score test is developed appropriate for the comparison of two groups with respect to disease development without need of any assumption concerning lethality of the disease concerned

global compactness properties of semilinear elliptic equations with critical exponential growth abstract sequences of positive solutions to semilinear elliptic equations of critical exponential growth in the plane either are precompact in the sobolev h 1 topology or concentrate at isolated points of the domain for energies allowing at most single point blow up we establish a universal blow up pattern near the concentration point and uniquely characterize the blow up energy in terms of a geometric limiting problem

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

quantization for a fourth order equation with critical exponential growth for concentrating solutions 0 u k rightharpoonup 0 weakly in h 2 to the equation delta 2 u k lambda k u k e 2u k 2 on a domain omega subset mathbb r 4 with navier boundary conditions the concentration energy lambda lim k rightarrow infty int omega delta u k 2 dx is shown to be strictly quantized in multiples of the number lambda 1 16 pi 2

a dose response test via closed form solutions for constrained mles in survival sacrifice experiments in most survival sacrifice experiments in animal carcinogenicity studies the onset of the tumour of interest is not clinically observable due to the complexity of constraints for a biological justification recently developed methods for estimating the tumour onset function and tumour specific survival function employ computer intensive numerical solutions in this paper closed form solutions for nonparametric maximum likelihood estimators are derived under explicit and implicit inequality constraints obtained from the monotonicity of the survival functions our methods do not require cause of death information the proposed methods can be used to estimate the tumour onset function and the survival function of the tumour of interest we use the proposed estimators for the development of our new dose response trend test a modification of the poly k test is made by replacing the time at risk weight to a function of the tumour onset survival function the weighted least square regression method is applied to the estimated survival functions in order to construct a dose response trend test a simulation study is conducted to evaluate the performance of the proposed test and compare it with existing trend tests a real example is used to illustrate the methods

tupling in three dimensional reversible mappings feigenbaum s parameter scaling exponents s are calculated for l 3m l 5m l 7m l 9m sequences of periodic orbits in a class of three dimensional non measure preserving reversible mappings some preliminary results on orbit scaling exponents for l 3m sequences are also included scaling exponents found so far are the same as those found for two dimensional reversible mappings

bifurcations and chaos in the ϕ4 theory on a lattice the authors have studied numerically and analytically the discrete phi 4 model defined by bak and pokrovsky see phys rev lett vol 47 no 13 p 958 61 1981 the model may describe peierls systems structural instabilities metal insulator transitions etc in solid state physics the phi 4 theory can be formulated as an area preserving two dimensional mapping this mapping exhibits infinite series of period doubling bifurcation leading to chaos the bifurcations are characterised by universal numbers delta 8 72109 and alpha 4 0180 which appear to be identical to those found by bountis 1981 for the henon mapping but different from the feigenbaum numbers for dissipative systems in addition novel features arise because of marginally stable fixed points and the splitting of one 2 cycle orbit into two 2 cycle orbits

critical behaviour and intermittency in sinai s billiard abstract we study numerically the transition that occurs in the dynamics of sinai s billard when the circles come into contact singular behaviours are predicted by a local analysis of the region of phase space where the motion is nearly integrable the link with intermittency is discussed we use a renormalization transformation and deduce the scaling laws we recover transposed in the language of critical phenomena certain predictions of the ergodic theory

efficient estimation of the lifetime and disease onset distribution summary we study efficient nonparametric maximum likelihood estimation of the distribution of onset and lifetime associated with an irreversible disease that is only detectable at sacrifice or death we show that if the onset distribution is continuous then estimation of the lifetime distribution cannot be improved by using current status information on the time till onset in this case the kaplan meier estimator of the lifetime distribution is asymptotically efficient the nonparametric maximum likelihood estimator tries to use current status information on the time till onset but it is asymptotically equivalent to the kaplan meier estimator and it is outperformed by the latter in simulations the nonparametric maximum likelihood estimator of the onset distribution is shown to be an iteratively reweighted least squares estimator which can be computed with the weighted pool adjacent violators algorithm we show that it is unnecessary to estimate the weights iteratively since an initial estimate cannot be improved this insight leads to a simple explicit estimator which improves on the nonparametric maximum likelihood estimator of the onset distribution the results are verified with simulations and a data analysis example is provided

on analytic perturbations of a family of feigenbaum like equations we prove existence of solutions 0 a of a family of feigenbaum like equations phi x 1 is an element of lambda phi phi lambda x is an element of x tau x 0 1 where epsilon is a small real number and tau is analytic and small on some complex neighborhood of 1 1 and real valued on r the family 0 1 appears in the context of period doubling renormalization for area preserving maps cf gaidashev and koch preprint 171 our proof is a development of ideas of h epstein cf epstein 1986 pi epstein 1988 in epstein 1989 4 adopted to deal with some significant complications that arise from the presence of the terms epsilon x tau x in eq 0 1 the method relies on a construction of novel a priori bounds for unimodal functions which turn out to be very tight we also obtain good bounds on the scaling parameter lambda a byproduct of the method is a new proof of the existence of a feigenbaum coullet tresser function

moser functions and fractional moser trudinger type inequalities abstract we improve the sharpness of some fractional moser trudinger type inequalities particularly those studied by lam lu and martinazzi as an application improving upon works of adimurthi and lakkis we prove the existence of weak solutions to the problem n 2 u u e b u 2 in 0 1 b 0 with dirichlet boundary condition for any domain in r n with finite measure here 1 is the first eigenvalue of n 2 on

a comparison of tumour incidence analyses applicable in single sacrifice animal experiments monte carlo methods are used to evaluate the operating characteristics of several trend tests for comparing incidence rates of occult tumours the candidate tests are applicable in long term animal experiments that have just one sacrifice time and no information on cause of death when survival decreases with dose and the tumour is lethal the tests that do not adjust for survival or that assume tumours are non lethal behave conservatively whereas the test that assumes tumours are rapidly lethal rejects the hypothesis of equal incidence rates too frequently the only test which consistently operates at the correct level is one that specifies a constant difference between the death rates for animals with and without the tumour surprisingly the test that specifies a constant ratio of these death rates often is conservative finally a test based on a simple modification of the unadjusted test which accounts for differential mortality by scaling down the size of the risk set performs reasonably well in many cases among the tests that operate at the proper level the constant risk difference test consistently exhibits high power across a wide range of situations

proportional hazards model for repeated measures with monotonic ordinal response in this paper we present a proportional hazards model to determine the effect of predictors on repeated measures data with ordinal response that is monotonic over time the model can handle both categorical and continuous predictors it naturally models the time spent in the response states the assumption of proportional hazards can be partially assessed if replicates individuals with identical covariate vectors are available censoring can be incorporated into the model although it does require additional modelling if the censoring is informative

tests for differences in tumor incidence based on animal carcinogenesis experiments abstract in animal carcinogenesis studies statistical methods that compare the production of occult tumors should compare tumor incidence rates in general the incidence rate cannot be identified even approximately without both survival and sacrifice information if information from frequent sacrifices is available however a function that approximates the incidence rate can be identified without assumptions about the relationship between tumor incidence and death some earlier proposals that based comparisons on tumor prevalence rates and other functions also compared tumor incidence rates under special conditions but these tests can be biased when the conditions are violated tests that compare functions approximating tumor incidence rates can be developed using survival sacrifice information but these tests may be seriously inefficient in special cases in which sacrifices are not necessary to identify the tumor incidence rate

bifurcations of lattice structures equilibrium configurations of a model for a crystal in one spatial dimension with anharmonic and up to third neighbour harmonic interactions are related to orbits in a two or four dimensional space under a nonlinear symplectic mapping in this way a connection is given with dynamical systems with two or three degrees of freedom depending on the parameters the orbits are smooth or stochastic their characterisation in terms of their fixed points and their ljapunov exponents is given the mapping shows a large number of bifurcations of which one can distinguish various types and these are discussed

nonparametric prevalence and mortality estimators for animal experiments with incomplete cause of death data abstract age specific prevalence and mortality estimators are important descriptors of disease development and the subsequent effects of a disease on longevity nonparametric maximum likelihood estimators for the prevalence and mortality functions are available for the case in which incidental and fatal occurrences of a disease can always be distinguished this article generalizes these methods to allow the role of a disease in causing death to be uncertain for a subset of the animals dying with the disease the proposed analysis makes no assumptions about the degree of disease lethality data from sacrificed animals can be incorporated easily although sacrifices are not necessary if a certain representativeness assumption holds no restrictions are imposed on the distribution of survival times but the prevalence function is held constant over time intervals for stabilization purposes variances are estimated by inverting the observed information matrix an em algorithm simplifies the analysis when the

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

the complete set of hamiltonian intermittency scaling behaviors renormalization group equations describing the phenomenon of intermittency in hamiltonian systems are presented all solutions satisfying certain physical constraints are obtained they are the complete set of simple singularities further considerations lead to precise predictions for scaling behavior at the onset of intermittency

nonparametric estimation of the distribution of time to onset for specific diseases in survival sacrifice experiments this paper concerns the analysis of an animal survival sacrifice experiment designed to investigate the incidence of a particular disease of interest the disease is assumed to be irreversible and detectable only at death for example by a necropsy each observation can be of one of three types i death caused by the disease ii death from a competing cause such as sacrifice with the disease present or iii death with the disease absent a two dimensional em algorithm is proposed for the nonparametric maximum likelihood estimation of the distributions of the time to onset and of the time to death from the disease these can be compared with nonparametric estimators recently proposed by kodell shaw and johnson 1982 biometrics 38 43 58 and by dinse and lagakos 1982 biometrics 38 921 932 a slight modification of the algorithm permits the construction of likelihood based interval estimates of quantiles of the distributions some extensions and generalizations are indicated

rigidity for infinitely renormalizable area preserving maps the period doubling cantor sets of strongly dissipative henon like maps with different average jacobian are not smoothly conjugated as was shown previously the jacobian rigidity conjecture says that the period doubling cantor sets of two dimensional henon like maps with the same average jacobian are smoothly conjugated this conjecture is true for average jacobian zero for example the one dimensional case the other extreme case is when the maps preserve area for example when the average jacobian is one indeed the main result presented here is that the period doubling cantor sets of area preserving maps in the universality class of the eckmann koch wittwer renormalization fixed point are smoothly conjugated

classification of solutions to the higher order liouville s equation on mathbb r 2m we classify the solutions to the equation mu 2m 1 e2mu on mathbb r 2m giving rise to a metric g e 2u g mathbb r 2m with finite total q curvature in terms of analytic and geometric properties the analytic conditions involve the growth rate of u and the asymptotic behaviour of u at infinity as a consequence we give a geometric characterization in terms of the scalar curvature of the metric e 2u g mathbb r 2m at infinity and we observe that the pull back of this metric to s2m via the stereographic projection can be extended to a smooth riemannian metric if and only if it is round

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

regression analysis of a disease onset distribution using diagnosis data we consider methods for estimating the effect of a covariate on a disease onset distribution when the observed data structure consists of right censored data on diagnosis times and current status data on onset times amongst individuals who have not yet been diagnosed dunson and baird 2001 biometrics 57 306 403 approached this problem using maximum likelihood under the assumption that the ratio of the diagnosis and onset distributions is monotonic nondecreasing as an alternative we propose a two step estimator an extension of the approach of van der laan jewell and petersen 1997 biometrika 84 539 554 in the single sample setting which is computationally much simpler and requires no assumptions on this ratio a simulation study is performed comparing estimates obtained from these two approaches as well as that from a standard current status analysis that ignores diagnosis data results indicate that the dunson and baird estimator outperforms the two step estimator when the monotonicity assumption holds but the reverse is true when the assumption fails the simple current status estimator loses only a small amount of precision in comparison to the two step procedure but requires monitoring time information for all individuals in the data that motivated this work a study of uterine fibroids and chemical exposure to dioxin the monotonicity assumption is seen to fail here the two step and current status estimators both show no significant association between the level of dioxin exposure and the hazard for onset of uterine fibroids the two step estimator of the relative hazard associated with increasing levels of exposure has the least estimated variance amongst the three estimators considered

birth of a new class of period doubling scaling behavior as a result of bifurcation in the renormalization equation it is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power and a map summarizing values of a function of the dynamical variable of the first subsystem undergoes a bifurcation in the course of increase of it occurs at c 1 984396 and results in a birth of the period 2 stationary solution of the rg equation at 2 this period 2 solution corresponds to the universal period doubling behavior discovered earlier and denoted as the c type criticality kuznetsov and sataev in phys lett a 162 236 242 1992 by combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period 2 solution in powers of c the developed approach resembles the e expansion in the phase transition theory in which a trivial stationary point of the rg transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices

nonparametric joint estimators for disease resistance and survival functions in survival sacrifice experiments the nonparametric estimators proposed in this paper simultaneously estimate the disease resistance function s t for the distribution of time to occurrence of disease and the survival function p t for the distribution of time to death caused by the disease the estimators are applicable to animal survival data when the disease of interest is irreversible and the cause of death of each animal can be determined diseases need not be classified as universally lethal or universally nonlethal and time to occurrence of the disease of interest need not be observable directly it is shown that the proposed estimators are maximum likelihood estimators under the condition that s t p t is monotonically decreasing application of the derived estimators is illustrated with experimental survival sacrifice data and results of a simulation study of the estimators performance are reported

period doubling in six dimensional symmetric volume preserving maps period doubling in three symmetrically coupled two dimensional area preserving maps is numerically studied it is found that there is a two dimensional manifold on which all period 2n n 0 1 2 orbits of a period doubling bifurcation sequence lie on this manifold the universality classes of period doubling are just the three classes for two symmetrically coupled two dimensional area preserving maps each characterised by its own feigenbaum constants reported by mao and helleman phys rev a vol 35 p 1847 1987 it is also reported that for three non symmetrically coupled two dimensional area preserving maps three maps which are fixed under the renormalisation operator have been found the relevant eigenvalues of perturbation around the fixed maps are again the same as those found for two non symmetrically coupled two dimensional area preserving maps by mao and greene phys rev a vol 35 p 3911 1987 the above mentioned numerical and renormalisation results for the six dimensional maps agree with each other

quantization for an elliptic equation of order 2m with critical exponential non linearity on a smoothly bounded domain omega subset mathbb r 2m we consider a sequence of positive solutions u k stackrel w rightharpoondown 0 in h m to the equation delta m u k lambda k u k e mu k 2 subject to dirichlet boundary conditions where 0 k 0 assuming that 0 lambda lim k to infty int limits omega u k delta m u k dx infty we prove that is an integer multiple of 1 2m 1 vol s 2m the total q curvature of the standard 2m dimensional sphere

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

universal power law tails for singularity dominated strong fluctuations integrals of the type i t identical to integral integral dxdy delta t h x y are considered where h is a function from an ensemble labelled by parameters i can represent circulation times of fluid particles in the plane orbital periods or semiclassical densities of states for one dimensional hamiltonian systems spectral densities for two dimensional crystals or the strength of a wave pulse produced by propagation of a deformed step discontinuity as h varies over the ensemble i develops strong fluctuations associated with legendre singularities for unrestricted functions h the probability distribution p i describing the fluctuations of i decays according to a universal i 9 law obtained by a scaling argument involving arnold s classification of catastrophes if h is only quadratic in y p varies as i 10 if the integral defining i is one dimensional p varies as i 3 if it is three dimensional i decays no faster than i 48 the probability distribution of i identical to mod delta i delta t mod decays as i 2

on the existence and uniqueness of the maximum likelihood estimate of a vector valued parameter in fixed size samples the maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the hessian matrix is negative definite whenever the gradient vector vanishes the condition of constancy on the boundary cannot be completely removed cf tarone and gruenhage 1975 the theory is illustrated with several examples

period doubling bifurcations and universality in conservative systems we have studied numerically a sequence of period doubling bifurcations of periodic orbits of henon s conservative two dimensional mapping in agreement with other recent studies we also find evidence that such sequences possess universality properties similar to the ones observed for dissipative systems the corresponding universal constants however have significantly different values in each case we suggest that a possible explanation for this discrepancy may be the fact that conservative systems owing to their measure preserving property do not become asymptotically one dimensional as dissipative systems do

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

estimating tumor incidence rates in animal carcinogenicity experiments tumor incidence is the primary measure of carcinogenesis this article focuses on estimating time dependent incidence rates in animal experiments with few sacrifices when the context of observation is known for none or all of the animals dying with the tumor of interest previous results are obtained under relaxed assumptions the link with existing semiparametric and nonparametric procedures based on latent failure times is exploited by using these methods to compute maximum likelihood estimates of the incidence rates without introducing latent random variables nonparametric estimators that are appropriate when all contexts of observation are known are generalized to the case in which the contexts of observation are unknown for a subset of the tumor bearing animals

current status and right censored data structures when observing a marker at the censoring time we study nonparametric estimation with two types of data structures in the first data structure n i i d copies of c n c are observed where n is a finite state counting process jumping at time variables of interest and c a random monitoring time in the second data structure n i i d copies of c t i t c n c t are observed where n is a counting process with a final jump at time t e g death this data structure includes observing right censored data on t and a marker variable at the censoring time in these data structures easy to compute estimators namely weighted pool adjacent violator estimators for the marginal distributions of the unobservable time variables and the kaplan meier estimator for the time t till the final observable event are available these estimators ignore seemingly important information in the data in this paper we prove that at many continuous data generating distributions the ad hoc estimators yield asymptotically efficient estimators of formula see text estimable parameters

universality in the transition to chaos of dissipative systems the period doubling bifurcation process for two dimensional transforms exhibits a new class of universality when a small dissipation is taken into account the effective jacobian is then defined as a function of both the dissipation and the rank n of the cascade cycle 2n numerical simulations of a simple mechanical system and numerical calculations on the henon mapping show that the decrement lies on a continuous curve as function of the effective jacobian a method using this result to understand experimental data is explained and a first order approximation of the renormalization process yields an analytic expression of the curve among the different transitions to chaos the period doubling bifurcation cascade 1 2 has been extensively studied this transition is characterized by an experimental convergence rate of the bifurcation threshold sequence to the accumulation point the threshold of chaos it is well known that the decrement of this bifurcation cascade can take different values each value corresponds to a specific class of systems which can be characterized by some general features of the system undergoing the transition 3 4 5 we are concerned here with the two values i 4 699 the decrement of the well known one dimensional transform with a quadratic maximum 2 and ii 8 721 the decrement of a two dimensional non dissipative transforms 3 these two classes of systems are generic in physics and the two values i and ii are therefore relevant values of the decrement however these two exponents stand for the infinite dissipation case and the conservative one thus leaving out the general physical situation of a finite dissipation only hints of the effect of a small dissipation in a two dimensional mapping have been given 6 before the work of zisook 7 a thorough study of the effect of dissipation is set forth here the first two sections deal with the physical model used to perform the numerical investigation and the experimental data thus obtained a study of the renormalization process enables to generalise the relation n j n j2 first given by zisook in 7 to all transforms where the jacobian does not depend on the linearization point in the phase space furthermore a first order approximation gives an excellent analytic expression of the universal function displaying the crossover of the decrement between ii and i

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