cycloergodic properties of discrete parameter nonstationary stochastic processes it is shown that a large class of nonstationary discrete parameter stochastic processes possess novel ergodic properties which are referred to as em cycloergodic properties specifically it is shown that periodic components of time varying probabilistic parameters can be consistently estimated from time averages on one sample path the cycloergodic theory developed herein extends and generalizes existing ergodic theory for asymptotically mean stationary and n stationary cyclostationary processes and is presented in both wide sense and strict sense contexts

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

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

renormalization for golden circles we prove that twist maps of the cylinder that are attracted by any fixed point of mackay s renormalization operator have a transitive invariant golden circle provided the fixed point satisfies a few simple purely topological conditions these conditions can be verified by finite precision arithmetics they are fulfilled for the simple fixed point and seem to be fulfilled for the critical fixed point taking existence and hyperbolicity of the critical fixed point for granted we conclude that the standard map has a critical invariant golden circle the induced map on the circle is topologically conjugate to a rigid rotation we can show that the conjugator is holder continuous moreover it is not differentiable on a dense set of points

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

two alternative philosophies for estimation of the parameters of time series two alternative philosophical frameworks for the two problems of estimating the time invariant or time variant autocorrelation function and its fourier transform for stationary and cyclostationary time series are compared one is based on the stochastic process model and the other is based on the nonstochastic time series model it is then explained that results on estimator bias and variance for these two problems couched within the stochastic process framework have analogs within the nonstochastic framework the bias and variance results for cyclostationary time series that are available within these two frameworks are then summarized

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

the cramer rao lower bound for directions of arrival of gaussian cyclostationary signals the cramer rao lower bound on the variance of estimates of directions of arrival signal strengths and carrier frequencies and phases of cyclostationary signals arriving at an array is considered numerical evaluations of the bound for the estimates of directions of arrival show that it can be orders of magnitude lower than the corresponding bound for stationary signals despite the great number of unknown parameters in the former case comparisons with the mean squared error of direction estimates obtained by the cyclic least squares algorithm show that the bound is not overly optimistic

nonparametric time series analysis for periodically correlated processes correlation functions of continuous time periodically correlated processes can be represented by a fourier series with coefficient functions it is shown that the usual estimator for stationary covariances formed from a single sample path of the process can be simply modified to provide a consistent in quadratic mean estimator for any of the coefficient functions resulting from the aforementioned representation it is shown that if the process is gaussian and b sub k tau is a fourier integral with respect to a density function g sub k lambda a two dimensional periodogram formed from a single sample function can be smoothed along a line of constant difference frequency to provide a consistent estimator for g sub k lambda this natural extension of the well known procedure for stationary processes provides a method for nonparametric spectral analysis of periodically correlated processes

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

on the number of solutions of polynomial congruences and thue equations has only finitely many solutions in integers x and yv in the first part of this paper we shall establish upper bounds for the number of solutions of 1 in coprime integers x and y under the assumption that the discriminant d f of f is nonzero for most integers h these bounds improve upon those obtained by bombieri and schmidt in 5 in the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of nagell 30 ore 32 sandor 33 and huxley 19 in fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is in general best possible in the second part we shall address the problem of finding forms f for which 1 has many solutions for arbitrarily large integers h finally we shall obtain upper bounds for the number of solutions of certain thue mahler and ramanujan nagell equations by appealing to estimates of evertse gy6ry stewart and tijdeman 17 for the number of solutions of s unit equations

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

consistent estimation of the cyclic autocorrelation the cyclic autocorrelation is often used to describe nonstationary random processes the authors investigate the conditions under which the cyclic autocorrelation can be estimated consistently in mean square for discrete time gaussian processes they extend and generalize results of hurd 1989 and refine results of boyles and gardner 1983 they derive necessary and sufficient conditions for consistency in mean square of an estimator which are in the form of a single sum of autocorrelation coefficients in the form of a double sum of autocorrelation coefficients in the bifrequency domain and in terms of the average spectrum they also discuss the rate of convergence for this estimator

fixed points of feigenbaum s type for the equationf p λx λf x existence and hyperbolicity of fixed points for the mapn p f x 1f p x withf p p fold iteration and f p 0 are given forp large these fixed points come close to being quadratic functions and our proof consists in controlling perturbation theory about quadratic functions

towards an existence proof of mackay s fixed point it is proved using a computer that there exist symmetric analytic twist maps u and t which satisfy the fixed point equations eqalign u btutb 1 crt btututb 1 cr here b is a diagonal 2 2 matrix if u and t commute then u t is either a fixed point or period three point for mackay s renormalization factor

on thue s equation and more precisely for the number of primitive solutions to 1 1 that is solutions in coprime integers x y the first important results on the number of solutions of thue s equation were obtained by c l siegel in the case r 3 and in the binomial case f x y a x r b y r in view of his results and comments we should attribute to siegel the problem to decide whether the number of primitive solutions to 1 1 for irreducible f of degree r 3 could be bounded by a function depending only on r and h but otherwise independent of f in 1983 siegel s question was answered in the affirmative by j h evertse 2 as a special case of his results he also treats equations in number fields evertse obtains the bound

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

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