dimension of the boundary of quasiconformal siegel disks we study quasiconformal siegel disks with critical points in their boundaries the main result asserts that every subarc of the boundary of the siegel disk has the hausdorff dimension strictly larger than 1 and that the boundary does not have a tangent at any point

functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

farey level separation in mode locking structure of circle mappings abstract we consider various characteristic scales both on the x axis and in the parameter space for critical and near critical circle mappings juxtaposition of different scales for the set of all rational mode lockings exposes the effect of separation into layers this layered structure is shown to reproduce farey tree levels of the corresponding rotation numbers the separation is studied numerically for maps with various orders of singularity and interpreted in terms of coefficients of rotation function expansion

herman s theory revisited we prove that a c 2 smooth orientation preserving circle diffeomorphism with rotation number in diophantine class d 0 1 is c 1 smoothly conjugate to a rigid rotation we also derive the most precise version of denjoy s inequality for such diffeomorphisms

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

critical points on the boundaries of siegel disks let be a polynomial map of the riemann sphere of degree at least two we prove that if has a siegel disk g on which the rotation number satisfies a diophantine condition then the boundary of g contains a critical point

numerical computation of the asymptotic size of the rotation domain for the arnold family abstract we consider the arnold tongue of the arnold family of circle maps associated to a fixed diophantine rotation number the corresponding maps of the family are analytically conjugate to a rigid rotation this conjugation is defined on a maximal complex strip of the circle and after a suitable scaling the size of this strip is given by an analytic function of the perturbative parameter the main purpose of this paper is to perform a numerical accurate computation of this function and of its taylor expansion this allows us to verify previous theoretical results the rotation numbers we select are quadratic irrationals mainly the golden mean by introducing a nonstandard extrapolation process especially suited for the problem we compute all the quantities required rotation numbers arnold tongues fourier and taylor coefficients with high precision

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

standardized time series l p norm variance estimators for simulations this paper studies a class of estimators for the variance parameter of a stationary stochastic process the estimators are based on lp norms of standardized time series and they generalize previously studied estimators due to schruben we show that the new estimators have some desirable properties they are asymptotically unbiased and have low asymptotic variance we also illustrate empirically the performance of the lp norm estimators on various stochastic processes

the brjuno function continuously estimates the size of quadratic siegel disks if a is an irrational number yoccoz defined the brjuno function by a ni 0 a0 a1 i i i an 1 log 1 an where a0 is the fractional part of a and an 1 is the fractional part of 1 an the numbers a such that a i are called the brjuno numbers the quadratic polynomial p a z iu e2i az z2 has an indifferent fixed point at the origin if p a is linearizable we let r a be the conformal radius of the siegel disk and we set r a 0 otherwise yoccoz y proved that a i if and only if r a 0 and that the restriction of a iu a log r a to the set of brjuno numbers is bounded from below by a universal constant in bc2 we proved that it is also bounded from above by a universal constant in fact marmi moussa and yoccoz mmy conjecture that this function extends to r as a hi older function of exponent 1 2 in this article we prove that there is a continuous extension to r

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

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

on the numerical computation of diophantine rotation numbers of analytic circle maps in this paper we present a numerical method to compute diophantine rotation numbers of circle maps with high accuracy we mainly focus on analytic circle diffeomorphisms but the method also works in the case of enough finite differentiability the keystone of the method is that under these conditions the map is conjugate to a rigid rotation of the circle moreover although it is not fully justified by our construction the method turns out to be quite efficient for computing rational rotation numbers we discuss the method through several numerical examples

critical circle maps near bifurcation we estimate harmonic scalings in the parameter space of a one parameter family of critical circle maps these estimates lead to the conclusion that the hausdorff dimension of the complement of the frequency locking set is less than 1 but not less than 1 3 moreover the rotation number is a holder continuous function of the parameter

non computable impressions of computable external rays of quadratic polynomials we discuss computability of impressions of prime ends of compact sets in particular we construct quadratic julia sets which possess explicitly described non computable impressions

differentiable circle maps with a flat interval we study weakly order preserving circle maps with a flat interval which are differentiable even on the boundary of the flat interval we obtain estimates on the lebesgue measure and the hausdorff dimension of the non wandering set also a sharp transition is found from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval

siegel discs herman rings and the arnold family we show that the rotation number of an analytically linearizable element of the arnold family fa b x x a bsin 27rx mod 1 a b c ir o b 1 27r satisfies the brjuno condition conversely for every brjuno rotation number there exists an analytically linearizable element of the arnold family along the way we prove the necessity of the brjuno condition for linearizability of p x d z az 1 z d d and e z azez a e27ia at 0 we also investigate the complex arnold family and classify its possible fatou components finally we show that the siegel discs of pa d and e are quasidiscs with a critical point on the boundary if the rotation number is of constant type

the attractor of renormalization and rigidity of towers of critical circle maps we demonstrate the existence of a global attractor with a cantor set structure for the renormalization of critical circle mappings the set is invariant under a generalized renormalization transformation whose action on is conjugate to the two sided shift with a countable alphabet

siegel disks with smooth boundaries we show the existence of angles r z such that the quadratic polynomial p z e2i z z2 has a siegel disk with c smooth boundary this result was first announced by r perez marco in 1993

all bounded type siegel disks of rational maps are quasi disks we prove that every bounded type siegel disk of a rational map must be a quasi disk with at least one critical point on its boundary this verifies douady sullivan s conjecture in the case of bounded type rotation numbers

area and hausdorff dimension of sierpinski carpet julia sets we prove the existence of rational maps whose julia sets are sierpinski carpets having positive area such rational maps can be constructed such that they either contain a cremer fixed point a siegel disk or are infinitely renormalizable we also construct some sierpinski carpet julia sets with zero area but with hausdorff dimension two moreover for any given number s in 1 2 we prove the existence of sierpinski carpet julia sets having hausdorff dimension exactly s

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

the brownian excursion area a numerical analysis abstract using an explicit expression for the laplace transform of the brownian excursion area generating function a numerical analysis gives moments density and distribution function an asymptotic formula is given for small argument

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

complete devil s staircase fractal dimension and universality of mode locking structure in the circle map it is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil s staircase at the onset of chaos the complementary set to the stability intervals is a cantor set of fractal dimension d 0 87 this exponent is found to be universal for a large class of functions

the parameter planes of λz m exp z for m 2 we consider the families of entire transcendental maps given by f m z zm exp z where m 2 all functions f m have a superattracting fixed point at z 0 and a critical point at z m in the parameter planes we focus on the capture zones i e values for which the critical point belongs to the basin of attraction of z 0 denoted by a 0 in particular we study the main capture zone parameter values for which the critical point lies in the immediate basin a 0 and prove that is bounded connected and simply connected all other capture zones are unbounded and simply connected for each parameter in the main capture zone a 0 consists of a single connected component with non locally connected boundary for all remaining values of a 0 is a quasidisk on a different approach we introduce some families of holomorphic maps of mathbb c which serve as a model for f m in the sense that they are related by means of quasiconformal surgery to f m

wandering domains in the iteration of entire functions fonctions entieres a domaines multiplement connexes de normalite ensembles de fatou julia pour des fonctions qui commuttent exemples de type d herman classes de fonctions entieres sans domaines d errance

a transcendental julia set of dimension 1 we construct a non polynomial entire function whose julia set has finite 1 dimensional spherical measure and hence hausdorff dimension 1 in 1975 baker proved the dimension of such a julia set must be at least 1 but whether this minimum could be attained has remained open until now our example also has packing dimension 1 and is the first transcendental julia set known to have packing dimension strictly less than 2 it is also the first example with a multiply connected wandering domain where the dynamics can be completely described

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

a decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

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

on the area under a continuous time brownian motion till its first passage time the area swept out under a one dimensional brownian motion till its first passage time is analysed using a fokker planck technique we obtain an exact expression for the area distribution for the zero drift case and provide various asymptotic results for the nonzero drift case emphasizing the critical nature of the behaviour in the limit of vanishing drift the results offer important insights into the asymptotic behaviour of a number of discrete models we also provide a succinct derivation for the distribution of the maximum displacement observed during a first passage

goodness of fit tests in models for life history data based on cumulative hazard rates to check the validity of an assumed parametric model for survival data one may compare t the nonparametric nelson aalen plot of the cumulative hazard rate with a t the estimated parametric cumulative hazard rate being for example the maximum likelihood estimator convergence in distribution of n t a t and more general processes is studied employing the general framework of counting processes which allows for quite general models for life history data and for quite general censoring schemes the results are applied to the construction of 2 type statistics for goodness of fit cramer von mises and kolmogorov smirnov type tests are presented in the case where the unknown parameter is one dimensional

on the julia set of a typical quadratic polynomial with a siegel disk let 0 1 be an irrational number with continued fraction expan sion a1 a 2 a 3 and consider the quadratic polynomial p z 7 e 2i z z 2 by performing a trans quasiconformal surgery on an associated blaschke product model we prove that if log an o p n a sn 1 then the julia set of p is locally connected and has lebesgue measure zero in particular it follows that for almost every 0 1 the quadratic p has a siegel disk whose boundary is a jordan curve passing through the critical point of p b y standard renormalization theory these results generalize to the quadratics which have siegel disks of higher periods

on siegel disks of a class of entire maps let f c c be an entire map of the form f z p z exp q z where p and q are polynomials of arbitrary degrees we allow the case q 0 building upon a method pioneered by m shishikura we show that if f has a siegel disk of bounded type rotation number centered at the origin then the boundary of this siegel disk is a quasicircle containing at least one critical point of f this unifies and generalizes several previously known results

scalings in circle maps i letf be a flat spot circle map with irrational rotation number located at the edges of the flat spot are non flat critical points s x axv v 1 first we define scalings associated with the closest returns of the orbit of the critical point under the assumption that these scalings go to zero we prove that the derivative of long iterates of the critical value can be expressed in the scalings the asymptotic behavior of the derivatives and the scalings can then be calculated we concentrate on the cases for which one can prove the above assumption in particular let one of the singularities be linear these maps arise for example as the lower bound of the non decreasing truncations of non invertible bimodal circle maps it follows that the derivatives grow at a sub exponential rate

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

asymptotics of the distribution of the integral of the positive part of the brownian bridge for large arguments the double laplace transform of the distribution function of the integral of the positive part of the brownian bridge was determined by m perman and j a wellner as well as the moments of this distribution the purpose of the present paper is to determine the asymptotics of this distribution for large values of the argument and the corresponding asymptotics of the moments

torus maps and the problem of a one dimensional optical resonator with a quasiperiodically moving wall abstract we study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically with d 2 incommensurate frequencies we show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d dimensional torus that preserves a foliation by irrational straight lines in particular the jacobian of this map has d 1 eigenvalues equal to 1 we present rigorous and numerical results about several dynamical features of such maps we also show how these dynamical features translate into properties for the field in the cavity in particular we show that when the torus map satisfies a kam theorem which happens for a cantor set of positive measure of parameters the energy of the electromagnetic field remains bounded when the torus map is in a resonant region which happens in open sets of parameters inside the gaps of the previous cantor set the energy grows exponentially

are there critical points on the boundaries of singular domains generalizing a result of e ghys we prove a general theorem that implies that if a rational functionf of the riemann sphere of degree 2 leaves invariant a singular domainc a disk or a ring on which the rotation number off satisfies a diophantine condition provided that on bar c f is injective then each boundary component ofc contains critical point off the injectivity condition is always satisfied for singular disks associated to linearizable periodic elliptic points off z z n a withne n 2 andae we also show that the singular disks associated to periodic elliptic points off z e az that satisfy a diophantine condition are unbounded in in the end of the paper we give a survey of the theory of iteration of entire functions of

on the scaling ratios for siegel disks the boundary of the siegel disk of a quadratic polynomial with an irrationally indifferent fixed point and the rotation number whose continued fraction expansion is preperiodic has been observed to be self similar with a certain scaling ratio the restriction of the dynamics of the quadratic polynomial to the boundary of the siegel disk is known to be quasisymmetrically conjugate to the rigid rotation with the same rotation number the geometry of this self similarity is universal for a large class of holomorphic maps a renormalization explanation of this universality has been proposed in the literature

bounded type siegel disks of a family of sine families abstract we prove that for any positive integer n there is a polynomial p z with degree n such that the entire function g z p sin z has a bounded type siegel disk bounded by a quasi circle in the plane which passes through at least one critical point of g in addition let 0 1 be an irrational number of bounded type we prove that for any integer k 0 the boundary of the siegel disk of g z 0 sin z e 2 i 1 s 2 k d s centered at the origin is a quasi circle passing through exactly two critical points 2 and 2 with multiplicity k 1 respectively

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

quadratic julia sets with positive area we recently proved the existence of quadratic polynomials having a julia set with positive lebesgue measure we present the ideas of the proof and the techniques involved

completeness and basis properties of complex exponentials this paper is concerned with what might be termed the fine structure of the completeness and basis properties of complex exponentials we give new criteria for two sequences to have the same excess in the sense of paley and wiener a result that illuminates and supplements a well known completeness criterion of levinson and new examples and counterexamples pertaining to riesz bases

on the distribution of the square integral of the brownian bridge smirnov obtained the distribution f for his 2 test in the form of a certain series f is identical to the distribution of the the brownian bridge in the l 2 norm smirnov kac and shepp determined the laplace stieltjes transform of f anderson and darling expressed f in terms of bessel functions in the present paper we compute the moments of f and their asymptotics obtain expansions of f and its density f in terms of the parabolic cylinder functions and laguerre functions and determine their asymptotics for the small and large values of the argument a novel derivation of expansions of smirnov and of anderson and darling is obtained

density of hyperbolicity for classes of real transcendental entire functions and circle maps we prove density of hyperbolicity in spaces of i real transcendental entire functions bounded on the real line whose singular set is finite and real and ii transcendental self maps of the punctured plane which preserve the circle and whose singular set apart from zero and infinity is contained in the circle in particular we prove density of hyperbolicity in the famous arnol d family of circle maps and its generalizations and solve a number of other open problems for these functions including three conjectures by de melo salom ao and vargas we also prove density of real hyperbolicity for certain families as in i but without the boundedness condition our results apply in particular when the functions in question have only finitely many critical points and asymptotic singularities or when there are no asymptotic values and the degree of critical points is uniformly bounded

theory of circle maps and the problem of one dimensional optical resonator with a periodically moving wall we consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions notably well known results in the theory of circle maps which we review briefly imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially even if these intervals are dense for typical motions of the reflecting boundary in the complement there is a positive measure set of parameters where the energy remains bounded

an extended fatou shishikura inequality and wandering branch continua for polynomials abstract let p be a polynomial of degree d with julia set j p let n be the number of non repelling cycles of p by the famous fatou shishikura inequality n d 1 the goal of the paper is to improve this bound the new count includes wandering collections of non pre critical branch continua i e collections of continua or points q i j p all of whose images are pairwise disjoint contain no critical points and contain the limit sets of eval q i 3 external rays also we relate individual cycles which are either non repelling or repelling with no periodic rays landing to individual critical points that are recurrent in a weak sense a weak version of the inequality reads n n irr i eval q i 2 d 1 where n irr counts repelling cycles with no periodic rays landing at points in the cycle q i form a wandering collection b c of non pre critical branch continua 1 if b c is non empty and 0 otherwise

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

singular measures in circle dynamics critical circle homeomorphisms have an invariant measure totally singular with respect to the lebesgue measure we prove that singularities of the invariant measure are of holder type the hausdorff dimension of the invariant measure is less than 1 but greater than 0

the scaling of arnol d tongues at degenerate fold bifurcations abstract we indicate that for families of differentiable homeomorphisms of the circle the scaling of the placing of arnol d tongues as well as of their widths is determined by the order of degeneracy of the fold bifurcation at which the scaling is tested we describe a mechanism involving a converging sequence of iterates of the homeomorphism

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

mating siegel quadratic polynomials 1 1 mating definitions and some history mating quadratic polynomials is a topological construction suggested by douady and hubbard do2 to partially parametrize quadratic rational maps of the riemann sphere by pairs of quadratic polynomials some results on matings of higher degree maps exist but we will not discuss them in this paper while there exist several presumably equivalent ways of describing the construction of mating the following approach is perhaps the most standard consider two monic quadratic polynomials fi and f2 whose filled julia sets k fi are locally connected for each fi let di denote the conformal isomorphism between the basin of infinity c k f and c ed with i 00 00 and v oo 1 these bottcher maps conijugate the polynomials to the squaring map

upper bound for the size of quadratic siegel disks if is an irrational number we let p n q n n 0 be the approximants given by its continued fraction expansion the bruno series b is defined as b alpha sum n geq 0 frac log q n 1 q n the quadratic polynomial p z e 2i z z 2 has an indifferent fixed point at the origin if p is linearizable we let r be the conformal radius of the siegel disk and we set r 0 otherwise yoccoz proved that if b then r 0 and p is not linearizable in this article we present a different proof and we show that there exists a constant c such that for all irrational number with b we have b alpha log r alpha c together with former results of yoccoz see y this proves the conjectured boundedness of b logr

a proof of the marmi moussa yoccoz conjecture for rotation numbers of high type marmi moussa and yoccoz conjectured that some error function upsilon related to the approximation of the size of siegel disks by some arithmetic function of the rotation number theta is a holder continuous function of theta with exponent 1 2 using the renormalization invariant class of inou and shishikura we prove this conjecture for the restriction of upsilon to a class of high type numbers

improved kam estimates for the siegel radius for the siegel center problem we explore the possibility of improving the kam estimates with a view to possible extensions to hamiltonian systems the use of a suitable norm and explicit perturbative computations allow estimates to within a factor 2 of the siegel radius for the quadratic map

completeness of systems of complex exponentials and the lambert w functions we study some of the properties of the solution system e i nt of the delay differential equation y t ay t 1 we first establish some general results on the stability of the completeness of exponential systems in l 2 and then show that the solution system above is always complete but is not an unconditional basis in l 2 1 2 1 2

the size of quadratic p adic linearization disks we find the exact radius of linearization disks at indifferent fixed points of quadratic maps in cp we also show that the radius is invariant under power series perturbations localizing all periodic orbits of these quadratic like maps we then show that periodic points are not the only obstruction for linearization in so doing we provide the first known examples in the dynamics of polynomials over cp where the boundary of the linearization disk does not contain any periodic point

on the excess of complex exponential systems in l2 a a abstract in this article we give new criteria for two complex sequences to have the same excess in the sense of paley and wiener in l 2 a a as a result we prove that given any positive integer q a real number 0 1 2 and complex numbers 0 0 n n q i log n q n 1 the exponential system t k e i t n k 0 1 q 1 n has excess 0 in l 2

rigidity of critical circle mappings ii we prove that two c3 critical circle maps with the same rotation number in a special set are c1 conjugate for some 0 provided their successive renormalizations converge together at an exponential rate in the c0 sense the set has full lebesgue measure and contains all rotation numbers of bounded type by contrast we also give examples of c critical circle maps with the same rotation number that are not c1 conjugate for any 0 the class of rotation numbers for which such examples exist contains diophantine numbers

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

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

onset of stochasticity for some one dimensional systems we study the appearance of stochastic dynamics in two situations 1 i for one parameter families of maps of the interval f undergoing an infinite sequence of period doubling bifurcations we consider 0 0 e beyond feigenbaum s parameter value 0 2 ii for a two parameter family of circle maps fb x x b2pisin2 x we consider b 1 1 e 0 1 in both cases we estimate the relative measure of parameter values corresponding to stochastic behavior

constructing locally connected non computable julia sets a locally connected quadratic siegel julia set has a simple explicit topological model such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution we constructively produce parameter values for siegel quadratics for which the julia sets are non computable yet locally connected

robust rigidity for circle diffeomorphisms with singularities we prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities their c 1 smooth equivalence follows from exponential convergence of those renormalizations as an easy corollary any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are c 1 smoothly conjugate