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

presentation functions fixed points and a theory of scaling function dynamics presentation functions provide the time ordered points of the forward dynamics of a system as successive inverse images they generally determine objects constructed on trees regular or otherwise and immediately determine a functional form of the transfer matrix of these systems presentation functions for regular binary trees determine the associated forward dynamics to be that of a period doubling fixed point they are generally parametrized by the trajectory scaling function of the dynamics in a natural way the requirement that the forward dynamics be smooth with a critical point determines a complete set of equations whose solution is the scaling function these equations are compatible with a dynamics in the space of scalings which is conjectured with numerical and intuitive support to possess its solution as a unique globally attracting fixed point it is argued that such dynamics is to be sought as a program for the solution of chaotic dynamics in the course of the exposition new information pertaining to universal mode locking is presented

continuous functions on totally ordered spaces that are compact in their order topologies abstract the totally ordered spaces that are compact and connected in their order topologies are characterized as images of lexicographic cubes the banach space of continuous functions on the lexicographic product of compact totally ordered spaces whose spaces of continuous functions have locally uniformly convex norms has an equivalent locally uniformly convex norm if and only if the product is countable the banach space of continuous functions on a compact totally ordered space always has an equivalent kadec norm those compact totally ordered spaces for which the banach space of continuous functions has an equivalent locally uniformly convex norm are characterized in terms of the bounded decreasing interval functions that can be defined on the intervals of the totally ordered spaces some examples are discussed in detail

the fixed point property for mappings admitting a center abstract we introduce a class of nonlinear continuous mappings in banach spaces which allow us to characterize the banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping later on we give an application to the existence of zeroes for certain kinds of accretive operators

information loss under coarse graining a geometric approach we use information geometry in which the local distance between models measures their distinguishability from data to quantify the flow of information under the renormalization group we show that information about relevant parameters is preserved with distances along relevant directions maintained under flow by contrast irrelevant parameters become less distinguishable under the flow with distances along irrelevant directions contracting according to renormalization group exponents we develop a covariant formalism to understand the contraction of the model manifold we then apply our tools to understand the emergence of the diffusion equation and more general statistical systems described by a free energy our results give an information theoretic justification of universality in terms of the flow of the model manifold under coarse graining

alur dual renormings of banach spaces we give a covering type characterization for the class of dual banach spaces with an equivalent alur dual norm

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

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

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

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

on weakly locally uniformly rotund banach spaces abstract we show that every normed space e with a weakly locally uniformly rotund norm has an equivalent locally uniformly rotund norm after obtaining a discrete network of the unit sphere s e for the weak topology we deduce that the space e must have a countable cover by sets of small local diameter which in turn implies the renorming conclusion this solves a question posed by deville godefroy haydon and zizler for a weakly uniformly rotund norm we prove that the unit sphere is always metrizable for the weak topology despite the fact that it may not have the kadec property moreover banach spaces having a countable cover by sets of small local diameter coincide with the descriptive banach spaces studied by hansell so we present here some new characterizations of them

maximum agreement subtree in a set of evolutionary trees metrics and efficient algorithms the maximum agreement subtree approach is one method of reconciling different evolutionary trees for the same set of species an agreement subtree enables choosing a subset of the species for whom the restricted subtree is equivalent under a suitable definition in all given evolutionary trees recently dynamic programming ideas were used to provide polynomial time algorithms for finding a maximum homeomorphic agreement subtree of two trees generalizing these methods to sets of more than two trees yields algorithms that are exponential in the number of trees unfortunately it turns out that in reality one is usually presented with more than two trees sometimes as many as thousands of trees in this paper we prove that the maximum homeomorphic agreement subtree problem is cal np complete for three trees with unbounded degrees we then show an approximation algorithm of time o kn5 for choosing the species that are not in a maximum agreement subtree of a set of k trees our approximation is guaranteed to provide a set that is no more than 4 times the optimum solution while the set of evolutionary trees may be large in practice the trees usually have very small degrees typically no larger than three we develop a new method for finding a maximum agreement subtree of k trees of which one has degree bounded by d this new method enables us to find a maximum agreement subtree in time o knd 1 n2d

the w sequence for circle maps and misbehaved itineraries the w sequence an ordering of maximal itineraries for circle maps is constructed by means of the farey transformation and the gluing operation the similarity in the dynamical behaviour of the maps is explicitly described in terms of itineraries a class of badly ordered itineraries termed misbehaved is defined with the language of symbolic sequences a misbehaved itinerary is shown to imply a rotation interval

bifurcations from an invariant circle for two parameter families of maps of the plane a computer assisted study we consider a two parameter family of maps of the plane to itself each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point for certain parameter values there is a hopf bifurcation to an invariant circle which is smooth for parameter values in a neighborhood of the bifurcation point however computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point this paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type

normality and modulability indices part ii convex cones in hilbert spaces this paper proposes and compares several ways of measuring the degree of normality of a convex cone contained in a normed space the dual concept of modulability is also considered other notions like solidity and sharpness are also analyzed from a quantitative point of view

average locally uniform rotundity and a class of nonlinear maps abstract we consider some topological characterizations of dual banach spaces that admit an equivalent dual average locally uniformly rotund norm and provide a criterion for such renorming which involves the class of slicely continuous maps

convex sets which are intersections of closed balls in this paper we aim to investigate different questions concerning the stability of the set m of all intersections of closed balls in a normed space we are mainly concerned with i the stability of m under the closure of the vector sums ii the stability under the addition of balls we prove that i and ii are different properties which have strong connections with the geometry of the space they have interest both in finite and infinite dimension in the former case there is a link with linear programming theory we also study two more stability properties related to the well known binary intersection property mazur sets and mazur spaces are introduced as a natural family satisfying i we prove that every two dimensional normed space is a mazur space a result which distinguishes dimension d 2 from dimension d 3 we also discuss the connections between mazur spaces and porosity

universal behaviour of two coupled circle maps symmetric coupling of two critical circle maps near the golden mean rotation number is considered on the basis of a renormalization group method the three universal types of interaction are found the theoretical scaling predictions are confirmed by numerical calculations

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension

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